KINEMATICS  OF  MACHINERY. 


A   BRIEF  TREATISE   ON   CONSTRAINED 
MOTIONS  OF  MACHINE  ELEMENTS. 


BT 

JOHN  H.  BARR,  M.S.,  M.M.E. 

Consulting  Engineer,  Union  Typewriter  Company; 

Formerly  Professor  of  Machine  Design,  Sibley  College,  Cornell  University; 
Member  of  the  American  Society  of  Mechanical  Engineers. 


REVISED    BY 

EDGAR    H.  WOOD,   M.M.E. 

Professor  of  Mechanics  of  Engineering,  Sibley  College,  Cornell  University. 


SECOND 


TOTAL   ISSUE',    EIGHT   THOUSAND,   ,   ,      > 


NEW    YORK: 

JOHN  WILEY  &  SONS. 
LONDON:    CHAPMAN  &  HALL,  LIMITED. 
1911 


Copyright,  1899,  1 91  it 

by 
JOHN  H.  BARR. 


THE  SCIENTIFIC  PRESS 

ROBERT   DRUMMONO   AND   COMPANY 

BROOKLYN,    N.   Y. 


<r  i  73 


Engineering 
Library 


PREFACE   TO   THE   SECOND   EDITION 


IN  revising  this  book  several  entirely  new  articles  have  been 
added  to  the  original  text,  notably  those  on  helical  gears,  and 
methods  of  gear  cutting.  A  number  of  articles  have  been  either 
wholly  or  partly  rewritten,  especially  those  dealing  with  instant 
centers,  velocity  and  acceleration  diagrams,  rolling  hyperboloids, 
involute  teeth,  and  epicyclic  trains.  Many  minor  errors  have  been 
corrected,  and  several  figures  have  been  redrawn. 

Professor  Leslie  D.  Hayes  has  given  valuable  assistance,  both 
in  reading  the  proof  and  in  the  work  of  revision. 

E.  H.  WOOD. 
ITHACA,  NEW  YORK, 
January,  1911. 

ill 


225252 


PREFACE   TO   THE   FIRST   EDITION. 


THIS  book  is  the  outgrowth  of  a  somewhat  smaller  treatise  which 
was  prepared  and  printed  by  the  writer  in  1894  for  the  use  of  the 
classes  in  mechanical  and  electrical  engineering  at  Sibley  College, 
Cornell  University. 

After  having  used  the  original  for  several  years,  it  was  decided 
to  issue  the  work  in  revised  form,  making  such  corrections  and 
changes  as  experience  suggested. 

The  present  volume  was  prepared  especially  to  bring  together, 
and  to  present  to  the  students  in  a  condensed  text-book,  those  prin- 
ciples and  methods  which  are  deemed  most  important  in  a  general 
course  on  Kinematics.  This  is  the  only  excuse  offered  for  another 
book  on  a  subject  about  which  so  much  has  been  written.  No  pre- 
tension is  made  to  originality  except  in  the  arrangement  and  manner 
of  presenting  a  few  subjects.  Neither  is  the  present  work  offered  as 
in  any  sense  a  complete  treatise  on  the  Kinematics  of  Machinery. 
The  treatment  of  many  topics  has  been  much  abridged;  particularly 
the  portion  relating  to  toothed  gearing,  a  subject  which  is  exhaus- 
tively treated  in  numerous  available  works.  On  the  other  hand, 
the  discussions  of  the  applications  of  such  important  conceptions 
as  instantaneous  centres,  velocity  diagrams,  etc.,  are  rather  fuller 
than  are  found  in  many  of  the  shorter  works  on  Mechanism. 

The  treatment  of  these  subjects  follows  closely  that  given  by 
Professor  Kennedy  in  his  admirable  work  on  the  Mechanics  of 
Machinery. 

It  is  believed  that  the  presentation  of  principles  and  methods, 
with  illustrations  of  their  applications,  is  the  proper  line  to  adopt 


vi  PREFACE. 

in  a  text-book  intended  for  a  short  general  course  on  such  a  subject 
as  Kinematics.  The  detailed  description  of  usual  forms,  and  the 
discussion  of  the  innumerable  considerations  with  which  the  expert 
in  any  line  must  be  familiar  are  to  be  sought  in  special  treatises. 

Messrs.  A.  T.  Bruegel,  D.  S.  Kimball,  and  W.  N.  Barnard,  all 
of  whom  have  given  instruction  in  the  course  to  which  it  applies, 
have  rendered  valuable  assistance  in  the  preparation  of  the  present 
book.  Mr.  Bruegel  contributed  most  of  the  problems,  which  were 
developed  during  his  six  years  as  instructor  in  Kinematics  at  Cor- 
nell University.  Professor  Kimball  kindly  wrote  the  articles  on 
"  Acceleration  Diagrams "  and  "  Epicyclic  Trains,"  and  he  and 
Mr.  Barnard  have  cooperated  in  other  ways  in  the  revision. 

Many  earlier  works  have  been  consulted  and  drawn  on  in  the  prep- 
aration of  the  present  book.  The  following,  especially,  should  be 
mentioned :  Principles  of  Mechanism,  by  Professor  Willis ;  Machin- 
ery and  Millwork,  by  Professor  Rankine ;  Kinematics  of  Machinery, 
by  Professor  Reuleaux ;  Mechanics  of  Machinery,  by  Professor  Ken- 
nedy; Kinematics,  by  Professor  MacCord;  Machine  Design,  by 
Professor  Unwin ;  Elementary  Mechanism,  by  Professors  Stahl  and 
Woods;  Teeth  of  Gears,  by  Mr.  George  B.  Grant;  A  Practical 
Treatise  on  Gearing  (Beale),  published  by  the  Brown  and  Sharpe 
Manufacturing  Company. 

The  writer  desires  to  acknowledge  his  obligations  to  all  who 
have  in  anyway  aided  in  the  preparation  of  this  little  book. 

JOHN  H.  BARK. 

ITHACA,  NEW  YORK, 
October  1899. 


CONTENTS. 


CHAPTER   I. 

PAGE 

FUNDAMENTAL  CONCEPTIONS  OF  MOTION.    THE  NATURE  OF  A  MACHINE      1 

CHAPTER  II. 
GENERAL  METHODS  OF  TRANSMITTING  MOTION  IN  MACHINES ,. . . .    37 

CHAPTER  III. 

PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.    FRICTIONAL  GEAR- 
ING     78 

CHAPTER   IV. 
OUTLINES  OF  GEAR-TEETH.    SYSTEMS  OF  TOOTH-GEARING 110 

CHAPTER  V. 
CAMS  AND  OTHER  DIRECT-CONTACT  MECHANISMS 169 

CHAPTER  VI. 

Li  NK  WORK 1 80 

CHAPTER  VII. 
WRAPPING-CONNECTORS.    BELTS,  ROPES,  AND  CHAINS 22 1 

CHAPTER  VIII. 
TRAINS  OF  MECHANISM 233 

PROBLEMS  AND  EXERCISES 249 

INDEX 257 

rii 


KINEMATICS  OF  MACHINERY. 


CHAPTER  I. 

FUNDAMENTAL  CONCEPTIONS  OF  MOTION.      THE    NATURE  OF  A 

MACHINE. 

1.  Motion  is  a  change  of  position;  and  it  is  measured  by  the 
space  traversed.    Time  is  not  involved  in  this  conception.    A  train, 
in  running  between  two  stations  fifty  miles  apart,  has  the  same 
motion,  whether  the  time  occupied  be  one,  two,  or  three  hours. 
The  motion  of  a  crank-pin  in  making  a  revolution  is  independent 
of  the  time  required , 

2.  Linear  Velocity,  or  simply  velocity,  is  the  rate  of  motion  of  a 
point  along  its  path  in  space.     It  is  a  function  of  both  space  and 
time,  and  is  measured  in  compound  units  of  these  fundamental 
quantities ;  as  feet  per  second,  feet  per  minute,  miles  per  hour,  etc. 

ds 

In  mathematical  terms,  velocity  =  v  =  ~-^f    in    which   s    =    the 

space  passed  over  in  the  time  t. 

If,  in  the  illustration  of  the  preceding  article,  the  time  of  the 
run  between  the  stations  is  one  hour,  the  train  has  an  average,  or 
mean,  linear  velocity  of  fifty  miles  per  hour;  if  the  time  be  two  and 
a  half  hours,  the  mean  velocity,  or  speed,  as  it  is  often  called,  is 
twenty  miles  per  hour,  etc.,  or  1760  ft.  per  min.,  or  29'  4*  per  sec. 

3.  Acceleration,  or  linear  acceleration,  is  the  rate  of  change  of 
velocity.     Acceleration  is  expressed  in  the  same  system  of  space-  and 
time-units  as  the  velocity  itself  (as  feet  and  seconds,  feet  and  min- 


2  Kf^SM4TIC8  Ofr  MACHINERY. 

utes,  miles  and  hours,  etc.);  but  acceleration  involves  one  space- 
factor  and   two   time-factors.     The   mathematical   expression  for 

dv       d*s 
acceleration  is  p  =  —=j-  =  -=— ,. 

If  a  velocity  is  uniformly  increased  from  10  feet  per  second  to 
18  feet  per  second,  the  change  of  velocity  is  8  feet  per  second.  If 
this  change  takes  place  in  2  seconds,  the  rate  of  change,  or  the 
acceleration,  is  4  feet  per  second  per  second,  or  4  foot-seconds  per 
second,  or  4  feet  per  square  second.  If  the  increase  of  velocity  is 
not  uniform,  the  mean  acceleration  is  4  feet  per  square  second  in 
the  above  illustration,  although  the  actual  increase  of  velocity  in 
any  one  second  is  not  necessarily  4  feet  per  second. 

4.  Uniform  and  Variable  Velocity. — If  the  motion  of  a  body  is 
uniform  (that  is,  if  all  equal  increments  of  space  are  traversed  in 
equal  increments  of  time)  the  velocity  is  uniform,  and  is  equal  to 
the  space  traversed  in  any  time  divided  by  that  time.  If  the  veloc- 
ity is  uniform,  the  acceleration  is  zero.  If  a  body  moves  120  feet 
in  10  seconds,  with  a  uniform  velocity,  the  velocity  is  12  feet  per 
second,  equivalent  to  720  feet  per  minute. 

If  the  velocity  is  not  uniform,  the  space  divided  by  the  time 
gives  only  the  mean  or  average  velocity,  and  the  velocity  may  vary 
between  the  widest  limits  during  the  motion.  If  the  law  of  the 
motion  is  known,  the  velocity  at  any  instant  may  be  determined 
from  the  space  and  time;  otherwise,  only  the  mean  velocity  can  be 
determined  from  these  data. 

The  velocity  of  a  body  may  vary  uniformly,  the  velocity  increas- 
ing or  decreasing  by  equal  increments  with  each  equal  increment  of 
time,  in  which  case  the  acceleration  is  constant;  or  it  may  vary 
according  to  any  other  law.  For  our  present  purposes  it  is  only 
necessary  to  discriminate  between  uniform,  or  constant,  and  vary- 
ing velocity. 

Although  the  velocity  may  be  constantly  changing,  it  is  cus- 
tomary to  speak  of  a  body  as  moving  at  a  certain  velocity,  as  25  feet 
per  second,  30  miles  per  hour,  etc. ;  and  such  expressions  are  per- 
fectly correct,  even  though  the  velocity  does  not  remain  constant 
for  a  single  instant.  For  example:  a  train  of  cars  in  getting  up 
speed  passes  through  every  velocity  from  zero  to  the  maximum 


CONCEPTIONS  OF  MOTION.     MATURE  OF  A  MACHINE.         3 

Telocity  attained;  at  a  certain  stage  the  velocity  may  be,  say,  ten 
miles  per  hour,  and  in  coming  to  rest  the  velocity  again  passes 
through  this  same  value.  Perhaps  the  train  does  not  maintain  this 
particular  velocity  for  a  single  foot;  yet,  for  the  instant,  it  is  said 
to  have  this  velocity;  meaning  that  if  it  continued  to  move  with  the 
Telocity  that  it  has  at  this  instant  it  would  move  10  miles  in  one 
hour. 

5.  Relative  and  Absolute  Motion. — All  known  motions  are  rela- 
tive, for  change  of  position  can  only  be  noted  with  reference  to 
objects  at  rest  (or  assumed  to  be  at  rest),  or  by  reference  to  objects 
the  motion  of  which  is  known  (or  assumed  to  be  known).  We 
know  of  no  body  absolutely  at  rest,  nor  do  we  even  know  the  abso- 
lute motion  of  any  body  in  the  universe. 

In  treating  of  the  motion  of  a  body,  only  its  change  of  position 
with  regard  to  some  other  body,  or  its  motion  relative  to  that  other 
body,  can  be  considered. 

In  ordinary  problems  of  terrestrial  mechanics  the  earth  is  taken 
as  the  standard  from  which  to  reckon,  and  a  body  which  does  not 
change  its  position  relative  to  the  earth  is  said  to  be  at  rest,  station- 
ary, or  fixed  ;  of  course  recognizing  that  it  partakes  of  the  motion 
which  the  earth  has  about  its  axis,  around  the  sun,  and  in  common 
with  the  sun  through  space. 

In  problems  of  machinery  the  motions  of  the  parts  are  usually 
most  conveniently  taken  with  reference  to  the  frame  of  the  machine 
as  a  standard.  In  "stationary"  or  "  fixed"  machines  this  is 
equivalent  to  referring  these  motions  to  the  earth,  for  the  frame 
has  no  appreciable  motion  relative  to  the  earth;  but  in  such  cases 
as  locomotives  and  marine  engines,  for  example,  the  parts  have  very 
different  motions  relative  to  the  frame  and  to  the  earth.  In  these 
latter  cases  we  are  usually  concerned  with  the  motion  of  the  parts 
relative  to  the  fram.e,  or  with  the  motion  of  the  machine  as  a  whole 
(including  everything  connected  with  it)  relative  to  the  earth. 

The  function  of  the  machine,  in  these  cases,  is  to  impart  motion, 
relative  to  the  earth,  to  the  attached  train  or  ship,  and  incidentally 
to  itself;  but  this  motion  of  the  entire  system,  and  the  motion  of 
the  parts,  as  members  of  a  machine,  may  generally  be  treated  as 
quite  distinct,  though  related,  problems.  A  marine  engine  can  be 


4:  KINEMATICS  OF  MACHINERY. 

studied  as  an  engine  just  as  a  mill  engine  can  be  treated,  without 
considering  the  application  of  the  energy  beyond  the  engine  itself. 

As  we  know  nothing  of  the  absolute  motion  of  a  body,  and  can 
only  know  its  motion  relative  to  other  objects,  it  can  have  as  many 
relative  motions  as  there  are  objects  with  which  to  compare  its 
changes  of  position. 

A  pair  of  locomotive  drivers,  for  example,  rotate  on  their  axi& 
relative  to  the  frame;  they  roll  along  the  rails  (each  point  tracing 
a  curve  of  the  cycloidal  class)  relative  to  the  rails  or  the  earth; 
they  rotate  about  the  axes  of  their  pins  relative  to  the  attached  side 
rods;  and  have  still  different  motions  relative  to  the  wheels  on  the 
other  axles,  to  the  piston,  etc. 

It  is  important  to  get  a  clear  conception  of  relative  motion,  for 
in  the  study  of  mechanism  the  treatment  may  often  be  much  sim- 
plified by  referring  a  motion  to  some  member  other  than  the  frame 
of  the  machine,  as  to  other  moving  parts. 

Throughout  this  work  it  will  frequently  happen  that  the  motion 
of  a  part  relative  to  some  other  moving  part  will  be  discussed ;  but 
it  is  to  be  understood,  unless  distinctly  indicated  to  the  contrary, 
that  the  word  "motion"  refers  to  the  change  of  position  relative 
to  the  frame.  Likewise,  when  a  member  is  said  to  be  at  rest,  fixed 
or  stationary,  it  is  to  be  understood  that  its  position  relative  to  the 
frame  remains  unchanged. 

Two  portions  of  a  rigid  body  can  have  no  motion  relative  to 
each  other;  for  a  change  of  the  relative  positions  of  such  parts 
involves  a  change  of  form,  and  this  is  not  consistent  with  the  con- 
ception of  a  rigid  body.  It  will  be  evident,  upon  brief  reflection, 
that  two  separate  bodies  which  have  no  relative  motion  could  be 
rigidly  joined  without  affecting  any  motions  that  they  may  have; 
for  as  they  do  not  change  their  position  relative  to  each  other  they 
must  have  identical  motions  relative  to  all  other  bodies,  and  may 
be  treated  as  parts  of  the  same  body  so  far  as  their  motions  are  con- 
cerned. 

Bodies  which  have  no  motion  relative  to  each  other  have  the  same 
motion  relative  to  any  other  body. 

The  converse  of  this  statement,  that  all  bodies  which  have  the 
same  motion  relative  to  another  body  have  no  motion  relative  to- 


CONCEPTIONS  OF  MOTION.     NATURE  OF  A  MACHINE.         5 

each  other,  is  not,  however,  generally  true.  Take  the  example  of 
the  locomotive  driving-wheels,  again;  each  set  of  wheels  has  the 
same  motion  relative  to  the  frame,  as  well  as  to  the  rails,  but  the 
wheels  on  the  different  axles  do,  nevertheless,  have  motions  relative 
to  each  other;  for  these  different  sets  of  wheels  could  not  be  rigidly 
fastened  together  as  one  piece  without  preventing  motion  relative  to 
the  frame. 

6.  Velocity  Ratio. — In  many  problems  of  machine  motions,  the 
actual  velocity  of  the  parts  is  not  of  so  much  importance  as  the 
ratio  of  the  velocities  of  two  or  more  parts.  In  another  class  of 
problems,  the  actual  velocity  (relative  to  the  earth,  or  other  stand- 
ard) must  be  treated.  The  present  work  is  concerned  very  largely 
with  the  former  class,  and  it  is  necessary  to  get  a  clear  conceptiou 
of  the  term  velocity  ratio.  This  may  perhaps  be  best  accomplished 
by  a  few  illustrations. 

Take,  as  an  example,  an  ordinary  simple  hand-windlass,  in 
which  a  rope  is  wrapped  around  a  drum  of  known  size,  and  a  crank 
of  given  radius  is  attached  to  the  axis  of  the  drum.  If  the  crank 
be  turned  through  one  complete  revolution,  the  load  attached  to  the 
rope  will  be  raised  a  height  equal  to  the  circumference  of  one  coil. 
For  any  number  of  turns  of  the  crank,  or  fractional  turns,  the  load 
will  be  raised  a  proportional  height ;  and  it  matters  not  whether  the 
crank  be  turned  fast  or  slowly,  the  ratio  of  its  motion,  and  of  its 
velocity,  to  that  of  the  load  is  the  same,  depending  entirely  upon 
the  proportions  of  the  device.  The  ratio  of  the  velocities,  or  the 
velocity  ratio,  is  independent  of  the  actual  velocities,  and  of  the 
forces  transmitted.  In  the  case  cited,  the  ratio  is  the  same  whether 
a  load  of  one  ton  be  hoisted  ten  feet  in  one  second,  or  one  pound  be 
hoisted  one  foot  in  one  minute.  The  same  point  is  illustrated  in 
the  action  of  most  of  the  common  machines.  In  an  ordinary  steam- 
engine,  for  every  revolution  of  the  crank  the  connected  parts  go 
through  certain  definite  motions;  while  the  time  of  one  such  revo- 
lution of  the  crank  may  be  a  tenth  of  a  second  or  ten  minutes,  all 
the  parts  (with  the  exception  of  such  parts  as  the  members  of  the 
governor,  to  be  mentioned  later)  go  through  the  same  relative 
changes  of  position ;  and  though  the  actual  velocities  with  which 


t>  KINEMATICS  OF  MACHINERY. 

such  changes  take  place  are  very  different  in  the  two  cases,  the  ratio 
of  these  velocities  remains  the  same. 

7.  Path, — A  point  in  changing  its  position  traces  a  line  called 
its  path. 

The  statements  in  the  preceding  articles  on  the  motions  and 
velocities  of  bodies  apply  equally  to  every  point  in  a  moving  body, 
whether  the  path  of  the  point  be  rectilinear  or  otherwise.  This  is 
consistent  with  the  definitions  of  motion  and  velocity;  for  these 
definitions  state  that  motion  is  measured  by  space  traversed  (not 
restricted  to  space  in  a  right  line),  and  that  velocity  is  the  rate  of 
motion. 

The  path  of  a  point  may  be  of  any  form  whatever,  in  a  plane 
or  in  space;  it  may  be  a  straight  or  curved  line  of  finite  length, 
along  which  the  point  moves  from  end  to  end,  reversing  its  direc- 
tion of  motion  at  either  end,  so  that  it  passes  any  particular  posi- 
tion first  in  one  direction  and  then  in  the  opposite  direction ;  it 
may  be  a  closed  curve  so  that,  unless  the  curve  crosses  itself,  suc- 
cessive passings  of  any  position  are  always  in  the  same  direction ; 
or  it  may  be  an  infinite  straight  or  curved  line,  the  point  never 
twice  occupying  the  same  position,  except  in  the  special  case  in 
which  the  curved  path  crosses  itself.  There  are  many  cases, 
however,  in  which  the  path  is  definite  and  limited  in  both 
form  and  extent,  and  nearly  all  motions  of  mechanisms  are  of 
this  class. 

8.  Cycle ;  Period ;  Phase. — In  most  mechanisms  the  members  go 
through  a  series  of  relative  motions,  at  the  end  of  which  they  occupy 
the  same  relative  positions  as  at  the  beginning. 

The  completion  of  such  a  series  of  relative  motions,  with  the  re- 
turn of  the  members  to  the  relative  positions  which  they  had  at  first, 
constitutes  a  cycle. 

In  the  ordinary  steam-engine,  for  example,  the  cycle  corresponds 
to  one  revolution  of  the  crank,  whatever  the  time  occupied  by  the 
revolution.  In  a  common  type  of  gas-engine,  the  cycle  corresponds 
to  two  complete  revolutions  of  the  crank,  for  the  four  strokes  of  the 
piston  during  these  two  revolution  are:  a  suction  stroke;  a  com- 
pression stroke ;  a  working  stroke  (impulse) ;  and  an  exhaust  stroke. 
The  valve-gear,  in  this  case,  is  so  arranged  that  valve,  piston,  etc., 


CONCEPTIONS  OF  MOTION.     NATURE  OF  A  MACHINE.         7 

only  return  to  their  initial  relative  positions  after  the  completion  of 
four  strokes  of  the  piston,  or  two  revolutions  of  the  crank. 

The  time  elapsing  during  a  cycle  is  called  the  period. 

The  simultaneous  positions  occupied  by  the  members,  at  any 
instant  during  the  cycle,  constitute  a  phase. 

9.  Continuous,  Reciprocating,  and  Intermittent  Motion. — If  the 
direction  of  motion  does  not  reverse,  the  motion  is  sometimes  said 
to  be  continuous  (using  the  word  somewhat  differently  ifcan-ifi  the 
strict  mathematical  sense,  in  which  all  motion  is  continuous). 

Motion  is  said  to  be  reciprocating  if  its  direction  reverses. 

Motion  is  called  intermittent  when  it  is  interrupted  by  intervals 
of  rest. 

Motion  in  a  closed  path  may  be  continuous,  reciprocating,  or 
intermittent;  and  it  may  vary  as  to  velocity  in  any  manner  what- 
soever. 

Motion  in  a  path  of  finite  extent,  not  forming  a  closed  figure, 
must  be  reciprocating,  and  may  or  may  not  be  intermittent. 

10.  Plane  Motion ;  Rotation,  Translation. — Of  the  great  num- 
ber of  motions  available  in  machinery,  a  very  large  proportion  are 
included   in  three   classes  of   comparatively  simple   nature,  viz. : 
Plane  Motion,  Helical  Motion,  and  Spherical  Motion. 

Plane  Motion  is  by  far  the  most  common,  and  it  is  the  simplest 
class  as  well. 

If  any  plane  section  of  a  body  moves  in  its  own  plane,  all  points 
in  this  section  move  in  this  plane,  and  all  points  outside  of  this  sec- 
tion move  in  planes  parallel  to  the  given  section.  Such  a  motion 
constitutes  a  plane  motion.  Any  point  in  a  body  having  plane 
motion  may  trace  any  path  in  its  plane;  but  all  points  similarly 
located  in  the  other  parallel  planes,  that  is  all  points  lying  in  a 
common  perpendicular  to  the  different  planes  of  motion,  have  paths 
of  identically  the  same  form.  Thus,  in  Figs.  1  or  2,  if  the  section 
shown  shaded  always  moves  in  its  own  plane,  the  successive  positions 
of  the  perpendicular  through  any  point  as  p  must  always  be  parallel, 
and  therefore  all  points  in  this  perpendicular  move  in  equal  paths. 

The  property  of  plane  motions,  just  discussed,  greatly  simplifies 
the  treatment  of  these  motions,  as  the  motion  of  one  point  (or  of  a 
set  of  points)  in  any  section  represents  the  motion  of  all  similar 


KINEMATICS  OF  MACHINERY. 


points  in  other  sections ;  or  the  motion  of  a  single  section  (a  plane 
figure)  in  its  own  plane  represents  the  motion  of  the  entire  body. 
This  can  be  extended  even  farther,  for  the  motion  of  a  point  not  in 
the  particular  plane  represented  can  be  replaced  by  that  of  its 
corresponding  point  on  that  plane  (its  projection  on  the  plane), 
and  thus  the  motion  of  a  single  plane  figure  represents  all  the 


Fig.  I  !A 


Fig.  2 


motions  of  all  the  points  in  the  body.  For  example,  the  motions 
of  the  points  p,  s,  and  q  in  Figs.  1  or  2,  are  in  equal  paths,  and 
the  motion  of  any  one  of  these  points  may  be  taken  to  represent 
that  of  any  other.  The  motion  of  an  engine  crank  and  of  the 
eccentric  can  be,  and  often  are,  conveniently  shown  together,  as 
if  actually  in  one  plane. 

In  case  of  all  other  than  plane  motions,  however,  it  is  necessary 
to  show  the  various  positions  of  the  members  by  two  or  more  pro- 
jections, or  by  some  equivalent  system,  if  it  is  desired  to  completely 
represent  the  motion. 

Plane  Motion  is  either  a  Rotation,  a  Translation,  or  a  motion 
which  can  be  reduced  to  a  combination  of  these.  The  reduction  of 
the  general  motion  to  a  combination  of  rotation  and  translation  is 
not  always  to  be  desired,  however,  and  such  motion  will  often  be 
treated  as  a  class  of  itself,  without  relation  to  the  simpler  and  more 
special  classes  to  which  it  can  be  reduced. 

If  a  body  moves,  as  in  Fig.  1,  so  that  all  points  travel  in  paral- 
lel planes  and  at  constant  distances  from  a  fixed  right  line,  it  has 
a  plane  motion  of  rotation.  Examples:  pulleys,  cranks,  levers, 
etc.  It  is  not  necessary  that  the  motion  be  continuous ;  the  rota- 
tion may  be  continuous,  reciprocating,  or  intermittent. 


CONCEPTIONS  OF  MOTION.    NATURE  OF  A  MACHINE.         9 

If  a  body  moves,  as  in  Fig.  2,  so  that  all  points  move  with  equal 
velocities  in  equal  paths,  the  motion  is  a  translation.  If  these 
paths  are  parallel  right  lines,  the  motion  is  a  rectilinear  translation. 
Examples :  the  carriage  of  a  lathe,  piston  or  cross-head  of  an  engine, 
platen  of  a  planer,  etc.  If,  however,  the  paths  of  all  the  different 
points  are  equal  curves,  the  motion  is  a  curvilinear  translation* 
Example :  the  side  rods  of  a  locomotive. 

Eectilinear  translation  is  always  to  be  understood  when  the  word 
translation  is  used  without  qualification. 

A  rectilinear  translation  may  be  treated  as  the  special  case  of 
rotation  in  which  the  distance  to  the  axis  is  infinity,  or  as  rotation 
in  a  circle  of  infinite  radius. 

It  has  been  shown  that  the  plane  motion  of  a  body  is  completely 
represented  by  the  motion  of  any  section  taken  in  a  plane  of 
motion,  or  by  the  change  of  position  of  a  plane  figure.  Two  points 
suffice  to  locate  a  figure  in  a  plane,  and  hence  the  plane  motion  of 
a  body  is  determined  by  the  motion  (successive  positions)  of  any 
two  of  its  points  not  in  the  same  perpendicular  to  the  plane  of 
motion.  In  the  general  case  of  motion  in  space,  the  motion  is 
determined  only  by  the  motions  of  at  least  three  points,  not  in  one 
right  line.  For  if  the  motions  in  space  of  two  points  are  known, 
the  body  may,  in  the  general  case,  have  a  motion  of  rotation  about 
the  line  connecting  these  two  points;  but  the  motion  of  a  third 
point,  outside  of  this  line,  deter- 
mines the  motion  of  the  body 
completely. 

In  general,  the  motion  of  a 
body  in  a  plane  may  be  reduced  to 
an  equivalent  rotation  and  a  trans- 
lation. Thus,  Fig.  3,  the  motion 
of  the  body  A,  which  is  complete- 
ly determined  by  the  motion  of 
two  points  such  as  a  and  #,  or  by 
the  motion  of  the  line  connecting 
these  points,  corresponds  to  a  change  of  position  from  A  to  A'. 
This  change  of  position  can  be  conceived  as  made  up  of  a  transla- 
tion, a-b  to  a'-V,  and  a  rotation,  about  I'  through  the  angle 


10 


KINEMATICS  OF  MACHINERY. 


a'  V  a".  Or,  the  rotation  can  be  conceived  to  take  place  first,  fol- 
lowed by  the  translation. 

As  this  motion  is  a  perfectly  general  case  of  plane  motion,  the 
same  reasoning  applies  to  all  such  cases,  no  matter  how  large  or 
how  small  the  motion  may  be. 

11.  Helical  Motion. — If  all  the  points  in  a  body  have  a  motion 
of  rotation  about  an  axis,  combined  with  a  translation  parallel  to 
that  axis,  the  motion  is  a  Helical  Motion  (see  Fig.  4).  In  nearly 
all  cases  the  helical  motions  met  with  in  machines  are  regular  hel- 
ical motions,  in  which  there  is  a  constant  relation  between  the 
rotation  and  the  translation;  that  is,  the  ratio  between  the  transla- 


|A 


ft* 


or 


Fi9.  4 


Fig.  5 


tion  component  and  the  angular  component  is  constant.  The  pitch 
of  the  helix  is  the  translation  along  the  axis  corresponding  to  one 
complete  rotation,  and  in  a  regular  helical  motion  the  pitch  is  con- 
stant. 

12.  Spherical  Motion. — If  the  motion  of  a  body  is  such  that 
all  points  in  the  body  remain  at  constant  distances  from  a  fixed 
point  (see  Fig.  5),  the  motion  is  spherical.     All  points  in  the  body 
move  in  the  surfaces  of  spheres,  having  the  fixed  point  for  a  com- 
mon centre. 

13.  Relation  between  Plane,  Helical,  and  Spherical  Motions.— 
If  the  translation  component  (pitch)  in  a  helical  motion  be  in- 
creased till  it  equals  infinity,  the  motion  reduces  to  a  plane  trans- 
lation.    On  the  other  hapd,  if  the  translation  component  be  re- 


CONCEPTIONS  OF  MOTION.     NATURE  OF  A  MACHINE.      11 


duced  to  zero,  the  motion  reduces  to  plane  rotation.  It  is  thus 
seen  that  both  of  the  limits  of  helical  motion  are  plane  motions, 
and  that  plane  motion  of  rotation  or  of  translation  may  be  treated 
as  special  cases  of  helical  motion. 

If  the  distance  from  the  fixed  point  to  the  moving  body  in  a 
spherical  motion  be  increased  to  infinity,  the  surfaces  of  the  spheres 
in  which  the  points  of  the  body  move  are  reduced  to  planes,  and 
we  thus  see  that  plane  motion  may  be  treated  as  a  special  case  of 
spherical  motion.  The  much  greater  frequency  of  plane  motion 
and  its  simplicity  makes  its  consideration,  in  practical  cases,  as  a 


Fig.  6 


Fig.  7 


Fig.  8 


Fig.  9 


special  form  of  these  more  complex  motions  undesirable,  though 
this  view  of  the  case  is  not  without  interest. 

Motions  more  complicated  than  the  classes  just  mentioned  are 
sometimes  met  with  in  machinery,  and  some  of  these  will  be  dis- 
cussed in  subsequent  articles;  but  they  are  comparatively  so 


12  KINEMATICS   OF  MACHINERY. 

few,  and  are  so  varied  in  character,  that  a  classification  of  them  is 
not  practicable.  Figs.  6,  7,  8,  and  9  show  practical  examples  of 
plane  rotation,  plane  translation,  helical  motion  (regular)  and 
spherical  motion  respectively. 

14.  Graphic  Representation  of  Velocity. — The  direction  and 
velocity  of  a  motion  may  be  represented  by  a  right  line,  the  direc- 
tion of  which  indicates  the  direction  of  the  motion,  while  its 
length  represents  the  velocity  to  some  convenient  scale. 

If,  for  instance,  it  is  desired  to  represent  the  velocities :  20  feet 
per  second,  35  feet  per  second,  55  feet  per  second,  and  40  feet  per 
second,  by  lines  on  a  drawing,  or  diagram,  a  scale  can  be  adopted 
which  will  give  convenient  lengths  (say  10  feet  per  second  to  the 
inch);  and,  to  this  scale,  these  velocities  will  be  represented  by 
lines  2  inches,  3.5  inches,  5.5  inches,  and  4  inches  long,  respec- 
tively. In  a  similar  way,  velocities  in  other  units,  as  feet  per 


pa- 


.  10  Fig.  II 


minute,  miles  per  hour,  etc.,  can  be  indicated  to  suitable  scales. 
The  velocity  of  the  point  p  (Fig.  10)  whicli  has  a  motion  of  300 
feet  per  minute  in  a  rectilinear  path,  is  represented  to  a  scale  of 
200  feet  per  minute  to  the  inch,  by  the  line  p-v ,  1.5  inches  long. 
If  the  motion  is  in  any  other  path  than  a  right  line,  the  velocity 
at  any  position  may  still  be  represented  by  a  straight  line  tangent 
'to  the  path  at  that  position,  for  the  direction  of  the  curved  path 
at  any  point  coincides  with  the  tangent  at  that  point. 

In  Fig.  11,  the  velocity  of  the  point  p,  moving  in  the  curved 
path  at  the  rate  of  45  fee'.;  per  second,  is  represented  to  a  scale  of 
40  feet  per  second  to  the  inch  by  the  line  p-v,  1.125  inches  long, 
lying  along  the  tangent  to  the  path  and  through  the  given  posi- 
tion of  p.  • 

This  graphic  representation  of  velocity  is  of  the  greatest 
importance,  as  it  makes  many  solutions  possible  on  the  drawing- 
board  without  the  use  of  calculations;  giving  the  results 


CONCEPTIONS  OF    MOTION.     NATURE  OF  A  MACHINE.     13 

required  directly  in  connection  with  the  regular  process  of 
designing,  and  permitting  the  easy  determinations  of  results 
that  could  only  be  arrived  at  otherwise  by  tedious  algebraic 
methods. 

As  to  the  accuracy  of  these  graphic  methods,  it  may  be  said  that 
they  are  as  close  as  can  be  used  in  a  drawing  itself;  so,  for  the  ordi- 
nary purposes  of  designing,  they  are  all  that  can  be  desired  in  this 
respect.  Furthermore,  the  graphic  method  has  the  advantage  of 
showing  a  number  of  connected  quantities  in  their  true  relation, 
appealing  to  the  mind  through  the  eye  much  more  effectively  than 
do  numerical  quantities.  A  limited  experience  with  such  problems 
as  follow  in  this  work  will  impress  upon  one  the  value  of  this 
method. 

15.  Newton's  Laws  of  Motion. — Starting  with  the  statement  of 
Newton's  Laws,  which  enunciate  fundamental  relations   between, 
force  and  motion,  and  with  the  familiar  Parallelogram  of  Forces, 
we  can  readily  develop  the  theory  of  the  very  important  subject  of 
Resolution  and  Composition  of  Motions. 

NEWTON'S  LAWS. 

I. — Any  material  point  acted  upon  by  no  force,  or  by  a  system 
of  balanced  forces,  maintains  its  condition  as  to  rest  or  motion;  if 
at  rest  it  remains  at  rest;  if  in  motion  it  .moves  uniformly  in  a 
right  line. 

II. — Any  material  point  acted  upon  by  a  single  force,  or  by  a 
system  of  unbalanced  forces,  has  an  acceleration  of  motion  pro- 
portional to,  and  in  the  direction  of  the  force,  or  the  resultant  of 
the  system  of  forces. 

III.  Action  and  reaction  are  equal,  opposite,  and  simultaneous. 

16.  Parallelogram  of  Forces. — The  resultant  of   two  or  more 
forces  applied  at  a  point  of  a  body  is  the  single  force  which,  if  ap- 
plied at  the  same  point,  will  have  the  same  effect  on  the  body,  as 
to  rest  or  motion,  as  the  given  forces  themselves.      These  forces 
which  act  together  are  called  components  of  the  single  force,  which 
is  equivalent  to  their  combined  action. 

Forces  may  be  represented  graphically,  in  a  similar  manner  to 
that  already  explained  in  connection  with  the  representation  of 


14  KINEMATICS  OF  MACHINERY. 

velocities;  the  direction  of  the  line  indicating  the  direction  of  the 
force,  and  the  length  of  the  line  representing  the  magnitude  of  the 
force.  If  two  forces,  acting  on  a  point,  are  represented  in  this 
way,  the  resultant  of  these  forces  is  similarly  represented  by  the 
diagonal  of  the  parallelogram  formed  on  the  components  as  sides. 
For  the  proof  of  this,  see  Mechanics  of  Engineering,  by  Professor 
I.  P.  Church,  page  4. 

This  proposition  can  be  extended  to  cover  the  case  of  any  num- 
ber of  forces  acting  at  a  point;  for  the  resultant  of  any  two  of  such 
a  system  of  forces  can  be  found,  then  the  resultant  of  this  first  re- 
sultant (which  exactly  replaces  the  two  original  forces),  and  an- 
other of  the  forces  can  next  be  found,  the  resultant  of  this  last 
resultant  and  another  component  can  then  be  found,  and  so  on  till 
all  of  the  original  forces  have  been  combined.  The  last  resultant 
is  the  resultant  of  the  system.  By  the  reverse  of  the  process  just 
outlined,  a  single  force  can  be  replaced  by  two  or  more  components. 

The  process  of  finding  the  resultant  of  several  forces  is  called 
the  Composition  of  Forces  ;  the  reverse  process  of  finding  the  com- 
ponents of  a  force  is  called  the  Resolution  of  Forces. 


P 
Fig.  12  ^-         -V>2        ^g.  13 

17.  Resolution  and  Composition  of  Motions  and  Velocities. — 

While  a  point  may  be  acted  on  by  any  number  of  forces  simulta- 
neously, it  can  have  but  one  motion  at  any  time.  This  motion  may, 
however,  he  considered  as  the  resultant  of  two  or  more  component 
motions,  in  the  same  way  that  any  force  may  be  considered  as  the 
resultant  of  two  or  more  forces. 

According  to  Newton's  second  law,  if  a  point  p  (Fig.  12), 
which  is  initially  at  rest,  is  acted  on  by  a  single  force  Fu  the  point 
will  move  in  the  direction  of  the  force.  The  velocity  at  any  instant 
may  be  represented  to  scale  by  Vi.  If  (Fig.  13)  a  second  force  F2 
acts  simultaneously  on  p,  the  motion  will  be  in  the  direction  of  the 


CONCEPTIONS  OF  MOTION.    NATURE  OF  A  MACHINE.      15 

resultant,  Fr,  of  Fl  and  F2.  This  motion  may  then  be  considered 
as  the  resultant  of  two  component  motions  in  the  directions  of 
the  respective  component  forces.  Similarly,  the  velocity  of  p 
may  be  considered  as  the  resultant  of  the  velocities  of  the  two 
component  motions.  Evidently  this  resultant  velocity  is  the 
diagonal  of  the  parallelogram  of  which  the  component  velocities 
are  adjacent  sides. 

From  the  preceding  discussion,  the  following  proposition  can 
be  drawn: 

PARALLELOGRAM  OF  VELOCITIES. 

If  two  component  velocities  of  a  point  be  represented,  to  scale, 
by  the  adjacent  sides  of  a  parallelogram,  the  diagonal  of  the  paral- 
lelogram will  represent  the  resultant  velocities  to  the  same  scale. 

Conversely,  a  velocity  represented  by  a  line,  to  scale,  may  be 
resolved  into  any  pair  of  component  velocities,  which  are  repre- 
sented to  the  same  scale  by  the  sides  of  a  parallelogram  of  which 
the  first  line  is  the  diagonal. 

As  in  the  case  of  forces,  the  reduction  of  more  than  two  com- 
ponent velocities  to  one  resultant  can  be  effected  by  an  extension 
of  the  above  principles. 

This  method  can  be  applied  to  any  number  of  velocities, 
whether  in  one  plane  or  otherwise.  In  Fig.  14  the  resultant  of 
v^  and  v2  =va',  the  resultant  of  va  and  v3  =  vy,  the  resultant  of  Vb 
and  v^  =  vr,  =  the  resultant  of  the  system. 

In  Fig.  15  the  resultant  of  v1  and  v2  =  ya;  resultant  of  va  and 
v3  —  vr. 

If  the  single  velocity  vr  (Figs.  13,  14,  or  15)  is  given,  it  can  be 
replaced  by  the  velocities  of  which  it  is  the  resultant;  for  they, 
combined,  are  its  equivalent. 

Determining  the  resultant  of  a  system  of  velocities  is  called 
Composition  of  Velocities;  finding  the  components  of  given  veloc- 
ities is  called  Resolution  of  Velocities. 

A  system  of  velocities  can  have  but  one  resultant;  but  a  given 
-velocity  can  have  an  infinite  number  of  sets  of  components.  The 


16 


KINEMATICS  OF  MACHINERY. 


velocity  v  (Fig.  16)  may  have  for  components  v,  and  #,;  v'  and 
va',  or  any  number  of  sets  of  components;  or  the  resolution  is  in- 
definite, because  an  infinite  number  of  parallelograms  can  be 
drawn  with  the  line  v  for  a  diagonal. 


Fig.  14 


Fig.  15 


If  we  know:  (a)  the  direction  of  both  components;  (£)  the  mag- 
nitude of  both ;  or  (c)  the  magnitude  and  direction  of  one,  there  is 
a  definite  resolution  [case  (b)  admits  of  a  double  solution].  For 
illustration  of  these  three  cases  see  Figs.  17,  18,  and  19,  respec- 
tively. 


Fig.  16 


-m. 


Fig.  17 


Case  (a).  The  given  velocity  p-v,  Fig.  17,  is  to  be  resolved  into 
components  in  the  directions  p-m  and  p-n. 

From  the  point  v  draw  line  v-Vi  parallel  to  p-n,  and  cutting  p-m 
in  Vi ;  also  from  point  v  draw  line  v-v*,  parallel  to  p-m,  cutting  p-n 
in  v2;  p-Vi  and  p-v2  are  adjacent  sides  of  a  parallelogram  meeting  in 
p,  and  p-v  is  the  diagonal  of  this  parallelogram  through  this  same 
point  p,  hence  the  velocities  represented  by  p-Vi  and  p-v2  are  the 
components  of  p-v  in  the  given  directions,  p-m  and  p-n.  It  is  evi- 


CONCEPTIONS  OF  MOTION.    NATURE  OF  A  MACHINE.      17 


dent  that  no  other  parallelogram  can  be  formed  on  p-v  as  a  diag- 
onal with  its  sides  in  these  given  directions. 

Case  (b) :  The  given  velocity  p-v,  Fig.  18,  is  to  be  resolved  into 
two  components  of  the  magnitudes  indicated  by  m  and  w,  direc- 
tions to  be  determined. 

With  radius  m  and  centre  p  draw  the  arc  m,-m/,  and  with  same 
radius  and  centre  at  v,  draw  the  arc  ma-rw3';  also  with  radius  n  and 
centre  at  v  draw  the  arc  n^nf ,  and  with  same  radius  and  centre  at 


Fig.  18 


P> 


draw  the  arc 


P 

Fig.  19 

this  gives  four   intersections. 


Connect 


these  intersections  with  p  by  the  lines  p-viy  p-v^  p-vj,  and  p-vj. 
By  drawing  lines  from  v  to  each  of  these  intersections  of  the  arcs, 
it  is  seen  that  two  parallelograms  are  formed  (p-v^v-v^  and  p-v/- 
v-v2f),  each  having  the  given  velocity  p-v  for  a  diagonal,  with  sides 
(p-Vi  and  p-Vt,  and  p-Vi  and  p-v2' ',  respectively)  equal  to  the  re- 
quired components ;  hence  there  are  two  solutions  to  this  case,  both 
satisfying  the  condition  that  the  velocity  p-v  be  resolved  into  two 
components  of  values  m  and  n. 

Case  (c)  Fig.  19:  The  given  velocity,  p-v,  is  to  be  resolved 
into  the  component  p-vl9  known  as  to  magnitude  and  direction, 
and  another  component,  entirely  unknown. 

Draw  a  line  from  v  to  vl9  also  draw  a  line  from  v  parallel  to 
p-vlt  then  draw  a  line  from  p  parallel  to  v-vl9  cutting  the  line  last 
drawn  in  va.  p-v^  is  the  required  component;  for  the  given  com- 
ponent p-vl  and  this  line  last  found  form  adjacent  sides  of  a 
parallelogram  with  p-v,  the  given  velocity,  as  a  diagonal. 


18  KINEMATICS  OF  MACHINERY. 

It  will  be  seen  from  the  preceding  discussion,  that  in  the  reso- 
lution of  a  velocity  into  two  components,  or  the  composition  of  two 
velocities  into  one  resultant,  that  there  are  six  elements  involved, 
viz. :  the  directions  and  magnitudes  of  three  velocities,  and  that  if 
four  of  these  elements  are  known  the  other  two  may  be  determined, 
(except  for  the  double  solution  of  case  6,  in  which  two  values  satis- 
fying the  conditions  are  obtained). 

The  first  case  (a)  is  by  far  the  most  common  in  practical  examples. 

18.  Angular  Velocity. — When  a  point  is  revolving  about  some 
axis,  permanently  or  temporarily,  it  is  frequently  convenient  to 
express  its  rate  of  motion  in  angular  rather  than  in  linear  measure. 
This  rate  of  motion  may  be  expressed  in  any  system  of  time  and 
angular  units,  as  revolutions  per  minute  or  per  second,  degrees  per 
second,  radians  per  minute,  etc.  In  many  practical  problems  the 
rate  of  angular  motion  of  a  member  is  most  conveniently  stated  in 
terms  of  revolutions  per  unit  of  time ;  but  in  analytical  expressions 
the  arc  passed  over  by  a  point  is  often  more  readily  measured  in 
other  units. 

The  radian  is  an  arc  of  a  length  equal  to  the  radius  r  ;  hence  there 

are —  %n  —  6.283  radians  to  a  circumference;  or  a  radian  is 

T 

equivalent  to  360°  -=-  6.283  =  57.3°,  nearly. 

If  a  revolving  point  makes  n  revolutions  about  its  axis  per  unit 
of  time  the  space  passed  over  in  time  unity,  or  its  linear  velocity, 
is  v  =  %7irn\  and  the  angle  traversed  in  the  same  time  will  be 

2  nrn 

GO  — =  %7rn  radians. 

r 

If  the  body  A  (Fig.  20)  is  revolving  about  the  axis  through  O 
(which  is  perpendicular  to  the  plane  of  the  paper),  at  the  rate  of  n 
revolutions  per  unit  of  time,  the  point  p,  at  a  distance  r  from  the 
axis,  has  a  linear  velocity  of  %nrn\  another  point  at  p',  at  a  dis- 
tance r'  from  the  axis,  has  at  the  same  time  a  linear  velocity1 
'Znr'n  ;  and  any  two  points  at  different  distances  from  the  axis 
have  different  linear  velocities  at  any  instant.  But  all  points  in 
the  same  rigid  body  when  revolving  about  an  axis  must  describe 
equal  angles  in  the  same  time,  and  the  angle  (or  arc)  is  being  de- 
scribed at  a  rate,  expressed  in  radians  by  %nn.  This  expression 


CONCEPTIONS  OF  MOTION.    NATURE  OF  A  MACHINE.      19 

for  the  rate  of  angular  motion  is  what  is  called  the  Angular  Veloc- 
ity of  the  body;  and  it  is  necessarily  the  same  at  any  instant  for  all 
points  in  the  same  rigid  body.  It 
will  be  noticed  that  the  only  varia- 
ble in  the  expression  for  angular 
velocity  as  just  derived  is  n,  the  num- 
ber of  revolutions  per  unit  of  time. 

Comparing  the  expression  for 
angular  velocity  with  that  for  the 
linear  velocity  of  a  revolving  body, 
it  is  seen  that  it  corresponds  with 
the  linear  velocity  of  a  point  in  the  body  at  a  distance  from 
the  axis  equal  to  unity  :  from  which  we  deduce  the  statement : 
The  angular  velocity  of  a  body  is  numerically  equal  to  the  linear 
velocity  of  a  point  in  the  same  body  at  unit  distant  from  the  axis. 

The  relation  between  the  linear  and  the  angular  velocity  of  a 
point  which  is  most  frequently  used  and  one  that  should  be  firmly 
fixed  in  the  memory,  is 


Fig.  20 


.          Linear  velocity 

Angular  velocity  = — J- 

Radius 


v 

or  oo  =  — . 
r 


If  a  point  revolves  about  a  fixed  centre  with  a  linear  velocity  of 
60  feet  per  second  (720  inches  per  second),  and  with  a  constant 
radius  of  18  inches  (1.5  feet),  its  angular  velocity  is 


or 


GO  =  — —  =  40  (radians  per  second), 

720 
GO  =•  -r-Q-  =  40  (radians  per  second). 


The  space-units  which  measure  the  radius  and  the  linear 
velocity  must  be  the  same,  and  the  angular  velocity  is  then  ex- 
pressed in  radians  per  second,  or  per  minute,  according  to  whether 
the  linear  velocity  time-units  are  seconds  or  minutes. 

Angular  velocity  may  be  constant,  or  it  may  vary,  uniformly  or 
otherwise.  If  the  radius  remains  constant,  as  in  a  body  rotating 
about  an  axis  to  which  it  is  rigidly  connected,  the  angular  velocity 


20  KINEMATICS  OF  MACHINERY. 

must  vary  just  as  the  linear  velocity  of  any  one  point  varies,  as 
is  seen  from  the  above  relation;  or  it  varies  directly  as  n. 

19.  Instantaneous  Motion,  Instant  Centre,  Instant  Axis. — The 
instantaneous  motion  of  a  point  is  its  motion  at  any  point  in  its 
path.  It  was  shown  in  Art.  14  that  the  direction  of  this  instan- 
taneous motion  is  along  a  tangent  to  the  path  at  the  position  of  the 
point.  Thus,  in  Fig.  21,  if  a  point  is  moving  in  the  path  m-n,  the  di- 

*  rection  and  velocity  of  its  instan- 
taneous motion  when  it  occupies 
the  position  p,  may  be  repre- 
sented by  the  line  p— v,  tangent 
to  m— n  at  p.  This  is  equally 
true  whether  the  point  is  mov- 
ing in  the  path  t-t',  a-b,  a'-b', 
a"-6",  or  in  any  path  whatever 
which  is  tangent  to  p-v  at  p. 
The  instantaneous  motion  of  a 

Fia.  21      N'  point  is  therefore  independent 

of  the  form  of  its  path.     Motion 

in  any  of  the  possible  paths  is  equivalent,  for  the  instant,  to 
rotation  about  c,  c',  or  c",  or  about  any  point  in  the  line  N-N', 
drawn  through  p  perpendicular  to  p-v,  for  the  path  of  a  point 
having  such  rotation  would  be  tangent  to  the  other  paths  at  p. 

An  Instantaneous  Centre  (called  more  briefly  an  Instant  Centre) 
af  any  plane  motion  of  a  body  is  a  point  about  which  the  body  may 
be  considered  as  rotating  at  any  instant  relative  to  another  body  in 
the  same  plane.  In  Fig.  22  let  p-v  and  p'-v'  represent  the  velocities 
of  the  instantaneous  motions  of  any  two  points,  p  and  p'  in  the 
rigid  body  A,  moving  in  the  plane  of  the  paper,  and  let  p— n 
and  pf-nr  be  perpendicular  to  p-v  and  p'-vf  at  p  and  p'  respect- 
ively. Then  the  instantaneous  motion  of  p  is  equivalent  to  rota- 
tion about  some  point  in  p-n  as  a  centre.  Likewise  the  motion 
of  p'  is  equivalent  to  rotation  about  some  point  in  p'-n'.  Since 
A  is  a  rigid  body,  p  and  p'  can  have  no  motion  relative  to  each 
other,  and  the  centre  of  rotation  of  p  must  also  be  the  centre  of 
rotation  of  p'.  The  point  0,  at  the  intersection  of  p-n  and  p'-n' y 
is  the  only  point  that  meets  this  requirement.  It  was  shown 


CONCEPTIONS  OF  MOTION.    NATURE  OF  A  MACHINE.     21 

in  Art.  10  that  the  plane  motion  of  a  body  is  determined  by 
the  motion  of  any  two  of  its  points  not  in  the  same  perpendicular 
to  the  plane  of  motion.  Therefore  the  instantaneous  motion  of 
A  is  equivalent  to  a  rotation  about  0  as  a  centre.  In  other 
words,  0  is  an  instant  centre  of  the  motion  of  A.  This  motion 
is  assumed  to  be  relative  to  the  paper  or  any  reference  body  in 
the  plane  of  the  paper.  If  the  material  of  A  and  the  reference 
body,  B,  are  assumed  to  be  extended  to  include  0,  Fig.  23,  a 
pin  could  be  put  through  0,  materially  connecting  A  and  B 
and  the  instantaneous  motion  of  A  with  reference  to  B  would 
:not  be  interfered  with. 

The  instant  centre  of  the  relative  motion  of  two  bodies  is  a  point 


Fig.  22 


Fig.  23 


at  which  they  have  no  relative  motion;  it  is  the  only  point  common 
to  the  two  bodies  for  the  instant. 

It  will  be  noted  that  the  points  p  and  p'  may  be  moving  in 
any  paths  whatever  so  long  as  these  paths  are  tangent  to  p-v 
and  p'—v'  at  p  and  p'  respectively.  Unless  these  paths  are  circular 
arcs  having  a  common  centre  at  0,  the  rotation  of  A  about  0 
does  not  continue  for  any  finite  time,  which  is  implied  when  0 
is  called  an  instant  centre.  If  the  paths  of  p  and  p'  are  such 
circular  arcs,  0  is  a  permanent  centre  as  well  as  an  instant 
centre. 

In  general,  it  is  only  necessary  to  know  the  direction  of  motion 
of  two  points  in  a  body  having  plane  motion,  in  order  to  deter- 
mine the  location  of  the  instant  centre.  When  the  points  are 
moving  in  parallel  paths  special  treatment  is  necessary.  If  the 
motions  are  at  right  angles  to  the  line  joining  the  two  points, 


22 


KINEMATICS  OF  MACHINERY. 


as  in  Fig.  24,  the  instant  centre  lies  somewhere  on  this  line. 
The  linear  velocities  of  points  in  a  rigid  body  being  propor- 
tional to  their  distances  from  the  centre  of  rotation,  the  loca- 
tion of  the  instant  centre,  0,  may  be  found  from  the  proportion 
p-v:p'-v' : :  0-p :  0-pf,  by  the  construction  indicated.  When,  as  in 
Fig.  25,  the  common  direction  of  motion  of  the  two  points  is  not  at 
right  angles  to  the  line  joining  them,  the  perpendiculars  p-n  and 
p'-nf  are  parallel,  and  their  intersection,  0,  is  at  infinity.  The 
instantaneous  motion  of  the  body  is  a  rotation  about  a  centre 
at  infinity,  or  it  is  translation.  In  this  case  all  points  of  the 
body  have  equal  linear  velocities. 

It  is  more  exact  to  refer  to  rotation  or  revolution  about  an 


n at  oo 


Fig.  24 


Fig.  25 


axis  than  about  a  centre.  In  the  case  of  plane  motion  the  axis 
of  rotation  is  always  perpendicular  to  the  plane  of  motion,  and 
pierces  every  section  of  the  body  parallel  to  the  plane  of  motion 
at  its  centre  of  rotation  (Fig.  1).  Since  the  motion  of  each 
section  completely  represents  the  motion  of  the  whole  body,  it 
is  customary  in  dealing  with  plane  motions  to  refer  to  instant 
centres  instead  of  the  corresponding  instant  axes. 

It  was  shown  in  Art.  10  that  the  motion  of  a  body  in  space 
is  determined  by  the  motion  of  any  three  of  its  points  not  in 
the  same  right  line.  Such  motion  is  at  any  instant  equivalent 
to  rotation  about  an  axis  combined  with  translation  parallel  to 
that  axis.  That  is,  it  is  a  form  of  helical  motion,  as  denned  in 
Art.  11.  In  Fig.  26,  the  instantaneous  velocities  of  any  three 
points  of  a  rigid  body  having  any  motion  whatever  in  space  are 


CONCEPTIONS  OP  MOTION.    NATURE  OF  A  MACHINE.    23 

represented  by  p-v,  p'-v',  and  pff-vff*  In  order  that  the  instan- 
taneous motion  of  the  body  may  be  equivalent  to  a  rotation 
about  some  such  axis  as  X—X',  combined  with  a  translation 
parallel  to  that  axis,  the  components  of  p—v,  p'—vf,  and  p"—v" 
perpendicular  to  X-X'  must  represent  a  plane  rotation  about 
X-X',  and  those  parallel  to  X-Xf  must  all  be  equal.  That  the 
axis  X-X'  can  be  located,  and  that  the  three  instantaneous 
velocities  can  be  resolved  into  such  components  is  shown  as  fol- 
lows: 

From  p0  in  Fig.  26 (a),  the  lines  p0-v0,  p0-v0',  and  p0~vo"  are 
drawn  respectively  parallel  and  equal  to  p-v,  p'-v',  and  p"-v" 


Fig.  26 

in  Fig.  26.  The  three  points  VQ,  VQ',  and  v0"  determine  a  plane. 
The  line  p0~so  is  drawn  perpendicular  to  this  plane,  piercing  it 
at  sQ.  The  projections  of  PQ—VO,  PQ—VQ,  and  PQ—VO"  on  p0~so  are 
all  equal  to  PO-SQ.  The  projections  of  the  same  lines  on  the 
plane  perpendicular  to  £>0~so  are  respectively  s0-v0,  s0-t>0',  and 
so~vo"'  These  are  all  perpendicular  to  p0~~so'-  I*1  Fig.  26  the 
components  of  p-v,  p'-v',  and  p"-v"  taken  parallel  to  p0~5o  are 
p— s,  p'—s',  and  p"—s".  These  are  all  equal  to  p0—s0  and  represent 

*  When  these  velocities  are  assumed  it  is  necessary  that  the  velocity  of  each 
point  be  so  taken  that  its  component  along  the  line  joining  the  point  to  each 
of  the  other  points  shall  be  equal  to  the  component  of  the  velocity  of  that 
point  along  the  same  line.  Otherwise  the  motion  would  change  the  distance 
between  the  points,  which  is  not  consistent  with  the  conception  of  a  rigid  body. 


24  KINEMATICS  OF  MACHINERY. 

a  translation  of  the  body  parallel  to  p0-s0.  The  components 
p-r,  p'-r',  and  p"-r"  are  respectively  parallel  and  equal  to  s0-^0, 
so~vo'>  so~V-  They  represent  plane  motion  of  the  body,  since 
all  are  parallel  to  the  plane  perpendicular  to  £>0~so-  Every  plane 
motion  of  a  body  has  been  shown  to  be  equivalent  to  an  instan- 
taneous rotation  about  an  axis  perpendicular  to  the  planes  of 
motion.  The  axis,  X—  X',  of  the  rotation  represented  by  p— r, 
p'-rf,  and  p"-r"  must  therefore  be  parallel  to  p0-sQ.  It  pierces 
every  plane  section  through  the  body  perpendicular  to  p0-sQ  in 
the  instant  centre  of  the  motion  of  that  section. 

The  motions  ordinarily  used  in  machinery  may  be  considered 
as  special  cases  of  space  motion.  When  the  axis  of  rotation  is 
fixed  and  there  is  a  constant  relation  between  the  rotation  and 
the  translation,  uniform  helical  motion  results.  When  the  trans- 
lation is  reduced  to  zero  and  the  axis  of  rotation  passes  through  a 
fixed  point,  the  motion  is  spherical.  Both  these  motions  may  be 
further  reduced  to  plane  motion  as  explained  in  Art.  13. 

20.  Free  and  Constrained  Motion. — It  follows  from  the  state- 
ments of  Art.  15  that  if  a  point  is  to  move  in  any  prescribed  path, 
the  resultant  of  all  forces  acting  upon  the  point  in  any  of  its  posi- 
tions must  lie  in  a  tangent  to  the  path  at  the  position  of  the 
point.*  If  the  path  be  other  than  a  straight  line,  this  involves  a 
constant  change  in  the  direction  of  the  resultant  force,  caused 
either  by  a  change  in  direction  or  magnitude  (or  both)  of  at 
least  one  of  the  components  of  this  resultant.  This  is  exactly 
what  takes  place  in  every  such  case;  but  the  method  of  this 
readjustment  of  the  resultant  force  affords  the  basis  of  a  very 
important  division  of  motions  into  two  classes,  viz.:  Free  and 
Constrained  Motions. 

A  body  which  has  no  material  connection  with  other  bodies  is 
called  a  free  body;  the  planets  are  examples  of  this  class  of 
bodies.  A  planet  revolves  around  the  sun  in  a  path  or  orbit 
determined  by  the  resultant  of  all  forces  acting  upon  it;  every 
disturbing  action  or  force  alters  its  path. 

The  motion  of  a  body  which  has  a  material  connection  with 
another  body,  permitting  motion  relative  to  that  body  only  in 

*  In  this  and  the  following  discussion  the  point  or  body  must  be  considered 
as  initially  at  rest  in  each  of  the  various  positions  it  occupies  in  tracing  its  path. 


CONCEPTIONS  OF  MOTION.     NATURE  OF  A  MACHINE.     25 


certain  restricted  paths,  is  said  to  be  constrained.  The  crank-pin  of 
an  engine  has  constrained  motion.  In  this  case,  if  motion  takes 
place  under  the  action  of  any  force  it  must  be  in  a  fixed  path,  and 
no  force,  whatever  its  direction,  short  of  one  that  will  break  or 
injure  the  machine,  can  cause  motion  in  any  other  path. 

The  primary  actuating  force  in  the  case  of  the  crank-pin  is  the 
pressure  or  pull  exerted  upon  the  pin  along  the  direction  of  the 
connecting-rod  (neglecting  frictional  influence).  This  primary 
force  does  not,  except  at  two  instants  in  each  revolution  of  the 
crank,  act  tangentially  to  the  path  of  the  body  acted  upon  ;  there- 
fore we  must  look  for  some  other  force  which  combined  with  this 
primary  force  gives  a  resultant  acting  tangentially  to  the  path  of 
the  crank-pin.  While  the  study  of  such  actions  is  not  strictly 
within  the  province  of  the  present  treatise,  it  is  important  to 
clearly  fix  the  nature  of  constrained  motion,  and  for  this  purpose 
the  distribution  of  force  acting  through  the  connecting-rod  upon 
the  crank-pin  of  the  ordinary  reciprocating  steam-engine  will  now  be 
briefly  considered.  Fig.  27  indicates  the  mechanism  of  the  engine 
(without  valve-gear).  Figs.  28,  29,  30,  31,  32,  33,  34  show  the 
connecting-rod  and  crank  in  different  positions,  or  phases.  The  full 
lines  indicate  the  directions  of  motion,  ana  UKJ  aasii-and-dot  lines 
indicate  forces  acting. 


Fig.  28 


In  Fig.  28  the  connecting-rod  is  at  right  angles  to  the  crank, 
and  therefore  its  centre  line  coincides  with  the  tangent  to  the  circle 
in  which  the  crank-pin  must  move.  As  the  force  P,  acting  on  the 
pin,  is  in  the  direction  of  the  centre  line  of  the  rod,  this  force  alone 
would  produce  motion  in  the  prescribed  path,  and  no  other  force 
need  be  considered  as  acting  to  produce  such  motion  at  this  par- 
ticular phase.  The  rod  is  under  compression. 

In  Fig.  29  the  condition  is  similar,  except  that  the  connecting- 
rod  is  now  under  tension  instead  of  compression,  and  the  action  on 


26  KINEMATICS  OF  MACHINERY. 

the  pin  is  a  pull  instead  of  a  thrust,  but,  as  before,  the  force  acts 
tangentially  to  the  path. 

In  Fig.  30,  however,  the  force  P'  exerted  by  the  connecting-rod 
on  the  pin  (thrust)  is  not  in  the  direction  of  the  tangent  to  the 
path,  and  hence  it  alone  cannot  produce  motion  in  the  required 


p' .. 


Fig.  29 


direction.  If,  however,  a  force  P",  be  introduced  in  the  direction 
of  the  centre  line  of  the  crank,  of  such  magnitude  that  it,  com- 
bined with  P1 ',  will  have  a  resultant  P  in  the  line  of  the  tangent  to- 
the  path  of  the  pin,  the  conditions  necessary  to  produce  motion  in 
the  required  direction  will  be  present ;  and  unless  such  a  com- 
ponent of  force  is  acting  in  conjunction  with  P',  the  required 
motion  cannot  take  place.  If  the  crank-pin  were  a  free  body  this- 
force  would  be  an  external  force,  but  it  will  be  seen  that  it  would 
be  very  difficult  to  apply  such  an  external  force  in  the  right  direc- 
tion and  of  the  proper  magnitude,  for  these  requirements  con- 
stantly change.  In  case  of  constrained  motion  the  material  con- 
nection (the  crank  in  this  case)  supplies  this  force  by  its  own 
resistance  to  a  change  of  form.  The  primary  acting  force,  alone, 
would  impart  motion  in  the  direction  of  its  own  line  of  action,  but 
this  motion  could  not  take  place  without  changing  the  form  of  the 
crank,  and  the  crank  offers  resistance  to  this  change,  by  just  the 
necessary  amount  for  constrainment.  As  action  and  reaction  are 
always  equal,  the  force  exerted  on  the  crank  to  change  its  form  is 
met  by  a  corresponding  counter-action,  or  reaction,  just  sufficient 
to  give  the  required  constraining  force,  and  to  cause  motion  in  the 
circle  of  which  the  crank  is  the  radius.  This  external  force  tend- 
ing to  change  the  form  of  the  crank  calls  out  within  the  material 
an  internal  molecular  action  known  as  stress,  and  this  action  is  just 
equal  to  the  external  force.  In  this  particular  phase  both  con- 
necting-rod and  crank  are  subjected  to  compression. 


CONCEPTIONS  OF  MOTION.     NATURE  OF  A  MACHINE.    27 


In  Fig.  31  the  condition  is  similar  to  that  of  the  preceding,  ex- 
cept that  the  connecting-rod  is  under  tension,  the  action  on  the  pin 


Fig.  31  Fig.  32 

is  a  pull,  and  the  resistance  of  the  crank  is  necessarily  reversed, 
the  stress  now  being  a  tension.  This  results  of  course  in  subjecting 
the  crank  to  tension  also,  and  as  it  is  of  a  material  that  will  resist 
this  action,  the  motion  in  the  required  path  is  secured  by  the 
combined  action  of  the  force  exerted  upon  the  pin  through  the 
connecting-rod  and  of  the  secondary  force  called  out  in  the  crank. 
Figs.  32  and  33  show  phases  in  which  the  stresses  in  the  connecting- 
rod  and  crank  are  not  similar. 

When  the  crank-pin  is  at  one  of  the  "dead  centres"  A  or  By 
as  in  Fig.  34,  it  will  be  noticed  that  the  force  exerted  by  the  con- 


Fig.  33 


Fig.  34 


necting-rod  is  at  right  angles  to  the  direction  of  the  pin's  motion, 
and  hence  no  force  combined  with  it  can  give  a  resultant  in  the 
direction  of  the  tangent  to  the  path ;  the  whole  effect  of  this  force, 
P',  is  now  to  compress  or  extend  the  crank  (change  its  form),  and 
none  of  it  is  available  in  moving  the  crank-pin.  If  it  were  not  for 
the  resistance  of  the  crank  at  this  time  the  pin  would  be  impelled 
in  the  direction  of  P',  at  right  angles  to  its  proper  path,  but  the 
resistance  of  the  crank  just  balances  the  force  received  from  the 
rod,  and,  according  to  Newton's  laws,  the  pin  is,  at  the  instant,  under 
a  system  of  balanced  forces,  and,  if  in  motion,  it  continues  to  move 
in  a  tangent  to  its  path,  unaffected  by  these  forces  except  as  they 


28  KINEMATICS  OF  MACHINERY. 

influence  friction.  Of  course  this  is  only  an  instantaneous  con- 
dition, and  therefore  the  pin  does  not  move  through  any  finite  dis- 
tance under  such  a  balanced  system  of  forces. 

Strictly  speaking,  the  condition  last  considered  is  not  equivalent 
to  the  action  of  no  force  at  all,  although  the  forces  are  balanced, 
for  the  pressure  of  the  pin  and  of  the  shaft  against  the  bearings 
results  in  a  frictional  resistance  tending  to  retard  the  motion. 
The  action  of  the  fly-wheel  also  modifies  the  motion  of  the  engine, 
reducing  the  fluctuation  of  velocity  that  would  be  experienced 
under  the  great  variation  of  the  resultant  force  throughout  the 
revolution;  but  neither  of  these  cases  need  be  treated  in  connec- 
tion with  the  present  discussion,  which  is  simply  intended  to  ex- 
emplify the  nature  of  constrained  motion. 

The  distinguishing  characteristic  of  a  constrained  motion  is  that, 
in  a  body  having  such  motion,  all  points  in  the  body  have  definite 
paths  in  which  they  move,  if  motion  takes  place  under  the  action 
of  any  force  whatever.  The  stresses  produced  in  the  restraining 
connections  supply  the  components  of  force  necessary  to  combine 
with  the  primary  force,  or  forces,  to  give  a  resultant  in  the  direc- 
tion of  the  prescribed  path.  If  these  connections  are  strong  enough 
to  resist  the  maximum  stress  to  which  they  are  thus  subjected,  no 
farther  attention  is  required  to  secure  the  proper  adjustment  of 
the  resultant  force  to  the  prescribed  path.  The  provision  of  the 
necessary  strength  is  in  the  province  of  another  branch  of  me- 
chanics, and  it  may  be  assumed  in  the  present  work  that  such 
strength  is  provided. 

It  will  be  seen  that  absolute  constrainment  is  not  possible 
by  the  ordinary  methods  employed  in  machinery  construction ; 
because  all  materials  are  somewhat  deformed  under  stress. 
Practical  constrainment  may  always  be  secured ;  that  is,  the  depart- 
ure from  the  desired  motion  can  be  reduced  to  any  required  limit. 
The  nature  of  the  constrainment  depends  upon  the  form  of 
the  constraining  members.  This  is  illustrated  in  Figs.  6,  7,  8 
and  9,  in  which  the  nature  of  the  relative  motion  of  the  parts  is 
plainly  determined  by  the  form  of  the  contact  surfaces.  The 
degree  of  constrainment  is  determined  by  the  dimensions  and 
material  of  the  constraining  members. 


CONCEPTIONS  OF  MOTION.     NATURE  OF  A  MACHINE.    29 

r 

All  motions  used  in  machinery  are  either  completely  or  partially 
Constrained. 

21.  Mechanics  is  the  science  which  treats  of  the  relative  motions 
and  of  the  forces  acting  between  bodies,  solid,  liquid,  or  gaseous. 

"  The  laws  or  first  principles  of  mechanics  are  the  same  for  all 
bodies,  celestial  or  terrestrial,  natural  or  artificial."  (Rankine.) 

22.  Mechanics  of  Machinery  treats  of  the  applications  of  those 
principles  of  pure  mechanics  involved  in  the  design,  construction, 
and  operation  of  machinery. 

Every  problem  of  mechanics  arising  in  connection  with  ma- 
chinery is  subject  to  the  laws  of  pure  mechanics,  and  we  could  con- 
ceive of  its  solution  by  the  general  methods  of  the  larger  science; 
but  the  operation  would  often  be  needlessly  difficult,  if  not  practi- 
cally impossible,  and  more  convenient  special  treatment  has  been 
developed  for  the  limited  class  of  phenomena  connected  with  prob- 
lems of  mechanism.  It  has  been  seen  that  constrained  motions 
are  much  more  easily  treated  than  are  free  motions;  and  all  problems 
of  motions  of  machines  are  included  under  constrained,  or  partially 
constrained,  motions.  It  is  mainly  the  distinction  between  free 
and  constrained  motions,  in  fact,  that  separates  the  Mechanics  of 
Machinery  from  the  more  general  science  of  Pure  Mechanics. 

23.  A  Mechanism,  or  train  of  mechanism,  is  a  combination  of 
resistant  bodies  for  transmitting  or  modifying  motion,  so  arranged 
that,  in  operation,   the  motion  of  any  member  involves  definite, 
relative,  constrained  motion  of  the  other  members. 

24.  A  Machine  consists  of  one  or  more  mechanisms  for  modify- 
ing energy  derived  from  natural  sources  and  adapting  it  to  the  per- 
formance  of   useful  work.     A   machine   may  consist   of   a  single 
mechanism  according  to  this  definition;  but  it  has  seemed  best  to 
make  the  following  distinction  between  a  mechanism  and  a  ma- 
chine: the  primary  function  of  the  former  is  to  modify  motion; 
while  that  of  the  latter  is  to  modify  energy,  and,  of  course,  inci- 
dentally motion.     The  term  "mechanism"  becomes  more  general, 
and  it  includes  the  elements  of  a  large  class  of  instruments  or  appa- 
ratus, such  as  clockwork,  engineers'  instruments,  models,  and  also 
most  forms  of  governors,  as  well  as  some  larger  constructions,  the 
function  of  which  is  essentially  the  modification  of  motion,  and 


30  KINEMATICS  OF  MACHINERY. 

which  only  do  work  incidentally,  such  as  the  overcoming  of  their 
own  frictional  resistance.  There  is  a  real  and  vital  distinction  be- 
tween machines  and  such  apparatus;  but  so  far  as  a  study  of  their 
motions  is  concerned,  no  such  distinction  need  usually  be  made. 

From  the  above  definitions  of  mechanisms  and  machines,  we 
may  derive  the  following: 

A  Machine  is  a  combination  of  resistant  bodies  for  modifying 
energy  and  doing  work,  the  members  of  which  are  so  arranged 
that,  in  operation,  the  motion  of  any  member  involves  definite, 
relative,  constrained  motion  of  the  others. 

The  essential  characteristics  of  a  machine  are: 

(a)  A  combination  of  bodies. 

(b)  The  members  are  resistant. 

(c)  Modification  of  energy  (force  and  motion)  and  the  perform- 
ance of  work. 

(d)  The  motions  of  the  members  are  constrained. 

(a)  A  machine  must  consist  of  a  combination  of  bodies. 

The  lever  does  not,  by  itself,  constitute  a  machine,  nor  even  a 
mechanism,  and  it  only  becomes  such  when  combined  with  the 
proper  fulcrum  or  bearing.  Without  this  complementary  member, 
properly  placed  and  sustained,  a  definite,  constrained  motion  is 
impossible. 

The  fulcrum  is  just  as  important  an  element  in  the  make-up  of 
the  machine  as  is  the  lever  itself.  The  screw  is  of  no  use  in  modi- 
fying motion  or  energy  unless  it  is  fitted  with  the  proper  envelope, 
usually  called  a  nut.  So  with  the  wheel  and  axle.  It  makes  no 
difference  whether  made  from  a  single  piece  of  material  or  built  up 
from  several  pieces  of  stock,  the  wheel  and  axle  is  essentially  one 
piece  when  completed,  as  there  is  no  relative  motion  between  the 
various  parts,  and  it  can  only  be  of  use  in  connection  with  appro- 
priate supports  or  bearings.  And  so  on  with  all  other  examples; 
the  simplest  machine  must  have  at  least  two  members,  between 
which  relative  motion  is  possible. 

(b)  The  members  of  a  machine  are  generally  rigid,  but  not  neces- 
sarily so.     Flexible  belts,  straps,  chains,  etc.,  confined  fluids  (liquid 


CONCEPTIONS  OF  MOTION.     NATURE  OF  A  MACHINE.    31 

or  gaseous),  and  springs,  often  form  important  parts  of  machines. 
The  flexible  bands  can  only  transmit  force  when  subjected  to  ten- 
sion; the  confined  fluids  transmit  force  only  under  compression; 
springs  may  act  under  tension,  compression,  torsion,  or  flexure. 
These  bodies  are  not  rigid,  in  the  usual  sense  of  the  word,  but  they 
are  resistant  under  the  particular  action  for  which  they  are 
adapted;  hence  they  can  be  used  in  special  applications  to  great 
advantage.  In  fact  their  value  in  such  applications  is  due  to  the 
absence  of  the  property  commonly  designated  as  rigidity. 

No  material  is  absolutely  rigid,  and  what  is  commonly  and  con- 
veniently called  a  rigid  body  is  one  in  which  the  distortions  under 
load  are  so  small  as  to  be  negligible  for  many  purposes. 

The  action  of  springs,  when  carefully  analyzed,  is  found  to  be 
identical  in  quality  with  that  of  the  so-called  rigid  bodies.  The 
characteristic  of  springs  is  the  magnitude  of  the  distortions.  Every 
solid  body  possesses  the  property  of  yielding  under  a  load  to.  a 
greater  or  less  degree,  following  the  same  general  law  as  springs, 
within  the  safe  working  limit  at  least.  The  difference  is  one  of 
degree  only,  but  in  this  difference  of  degree  lies  the  special  fitness 
of  springs  for  certain  parts  of  machines. 

(c)  The  machine  is  used  to  modify  energy  and  do  work. 

It  is  interposed  between  some  source  of  energy  and  the  work  to 
be  done,  and  it  adapts  this  energy,  as  supplied  by  or  derived  from 
natural  sources,  to  the  required  work. 

The  conception  of  a  machine  involves  the  conception  of  some 
source  of  energy,  an  effect  to  be  produced,  and  a  train  of  mecha- 
nism suitably  arranged  to  receive,  modify  and  apply  the  energy 
derived  from  this  source  to  the  desired  end. 

The  nature  of  the  source  of  energy  and  of  the  work  to  be  done 
determine  the  character  of  the  machine,  and  the  forms  of  the 
members  for  receiving  the  energy,  transmitting,  modifying,  and 
applying  it.  The  primary  natural  source  of  energy  may  be  the 
muscular  effort  of  animals,  wind,  water,  heat  (acting  through  sucL 
vehicles  as  steam,  air,  or  other  gases),  etc.  The  secondary  sources 
may  be  pulleys,  gears,  shafts,  etc.,  deriving  their  energy,  directly 
or  indirectly,  from  some  of  the  primary  sources.  The  prime 
movers — windmills,  water-wheels,  heat-engines,  etc. — are  driven 


32  KINEMATICS  OF  MACHINERY. 

from  primary  sources,  while  machinery  of  transmission — machine 
tools,  dynamos,  etc. — are  actuated  from  secondary  sources. 

In  a  machine-shop,  for  example,  the  source  of  energy  of  the 
tools  is  the  line-shaft,  or  the  counter-shaft,  according  as  the  latter 
is,  or  is  not,  treated  as  a  part  of  the  tool;  it  evidently  makes  no 
difference  how  this  shaft  is  driven,  so  far  as  the  study  of  the  indi- 
vidual tool  is  concerned.  The  source  of  energy  being  a  rotating- 
shaft,  the  member  of  the  machine  receiving  the  energy  must  be  a 
pulley,  gear,  sheave  or  other  form  capable  of  connection  with  such 
rotating-shaft.  Energy  may  be  transmitted  by  compressed  air  or 
by  water  under  pressure ;  then  the  receiving  member  may  be  a  Dis- 
ton,  reciprocating  in  a  suitable  cylinder,  or  a  wheel  with  appropri- 
ate vanes  or  blades  attached. 

In  a  similar  way  the  desired  result  determines  the  motions  and 
forms  of  the  members  producing  it.  When  metal  is  to  be  planed 
a  reciprocating  motion  is  usually  imparted  to  the  member  to 
which  the  piece  operated  upon  is  attached,  or  to  the  cutting  tool. 
Thus,  different  classes  of  work,  such  as  grinding  grains  or  min- 
erals, pumping  water  or  other  fluids,  compressing  air  or  other 
gases,  weaving  or  spinning,  cutting  woods,  stones,  or  metals,  the 
transportation  of  materials,  etc.,  each  require  an  appropriate 
modification  of  the  energy  imparted  to  the  receiving  member  of 
the  machine. 

In  general,  any  of  the  sources  of  energy  may  be  applied  to  pro- 
duce any  mechanical  effect  by  means  of  proper  trains  of  mechan- 
ism; and  this  gives  rise  to  a  very  great  number  of  possible  ma- 
chines. The  working  members  of  machines  have  been  classified  by 
Willis  as: 

(a)  Parts  receiving  the  energy. 

(b)  Parts  transmitting  and  modifying  the  energy. 

(c)  Parts  performing  the  required  work. 
To  these  might  be  added : 

(d)  Auxiliary  parts,  as  regulators,  etc. 

(e)  Frames   for  restraining  the   motions  and   sustaining  the 
machines. 

Various  classifications  of  the  parts  of  machines  have  been  made 


CONCEPTIONS  OF  MOTION.    NATURE  OF  A  MACHINE.     33 

by  different  writers,  but  that  of  Willis  has  perhaps  been  most  gen- 
erally accepted.  From  a  kinematic  standpoint,  such  classifications 
are  of  doubtful  value,  and  Reuleaux's  masterly  treatment  of  the 
subject  indicates  that  all  such  divisions  are  artificial  and  arbitrary. 
This  will  be  more  fully  discussed  under  Inversion  of  Mechanisms. 

The  working,  or  moving,  members  of  a  machine  may  be  levers, 
arms,  beams,  cranks,  cams,  wheels  with  treads,  blades,  vanes,  or 
buckets,  with  teeth  or  with  flat  or  grooved  rims,  etc. ,  screws  and 
nuts,  rods,  shafts,  links,  and  other  rigid  members;  as  well  as 
belts,  bands,  ropes,  chains  (flexible  members);  and  occasionally 
confined  fluids,  as  water,  oil,  air,  etc.  Many  modifications  of 
these  are  used,  and  an  indefinite  variety  of  forms  result,  yet,  kine- 
matically,  when  reduced  to  the  simplest  forms,  the  variety  of  mech- 
anisms is  much  less  than  would  at  first  appear. 

The  frames  which  support  the  working  parts  and  determine 
their  motions  are  almost  as  varied  in  form  and  materials  as  the 
moving  members  themselves,  but  are  capable  of  similar  simple 
treatment.  In  fact,  as  will  appear  later,  the  frame  may  be 
treated  as  exactly  equivalent  to  any  other  member  of  the  machine, 
and  so  far  as  relative  motion  of  the  members  is  concerned,  it  mat- 
ters not  which  particular  piece  is  made  "  stationary." 

The  leading  distinction  between  a  machine  and  a  "structure" 
(such  as  a  bridge)  is  that  the  former  serves  to  modify  and  transmit 
energy,  or  force  and  motion;  while  the  latter  modifies  and  trans- 
mits force  only.  Some  parts  of  machines,  as  the  fixed  frames, 
are  properly  structures,  while  as  a  whole  the  construction  is  a 
machine. 

(d)  The  relative  motions  of  the  members  of  a  machine  are 
constrained,  or  restricted  to  certain  definite  predetermined  paths, 
in  which  they  must  move,  if  they  move  at  all  relative  to  each, 
other. 

The  nature  of  constrained  motion  has  been  considered  in  Art.  20. 

The  leading  characteristic  of  a  mechanism  or  a  machine  is  the 
constrainment  of  its  motion.  A  structure  does  not  permit  relative 
motion  of  its  members,  or  at  most  it  only  allows  the  very  limited 
incidental  motions,  due  to  deformation  of  its  members  under  loads, 
the  effect  of  changes  of  temperature,  etc.  Occasionally  what  are 


34  KINEMATICS  OF  MACHINERY. 

usually  classed  as  structures,  or  parts  of  structures,  do  have  pre- 
scribed motions,  as  the  draw  of  a  bridge,  or  a  turn-table.  These 
are  not  properly  machines,  but  they  do  come  under  the  preceding 
definition  of  a  mechanism. 

All  artificial  combinations  of  bodies,  to  which  may  be  given  the 
general  name  of  constructions,  and  which  have  constrained  motion, 
may  be  classed  as  mechanisms;  and  if  intended  to  be  employed  in 
the  performance  of  useful  work,  as  machines. 

The  constrainment  in  mechanisms  is  sometimes 'partial,  or  in- 
complete. Thus  in  the  case  of  a  crane,  in  which  the  load  is  sus- 
pended by  a  chain  or  cable,  the  slightest  horizontal  force  will  sway 
the  load-hook,  and  therefore  change  its  path  from  the  right  line 
perpendicular  to  the  earth's  surface  in  which  it  normally  moves. 
This  does  not  affect  the  useful  operation  of  the  crane,  as  the  hook 
is  constrained  against  all  undesirable  motion,  and  for  practical  pur- 
poses the  action  is  just  as  good  as  if  the  constrainment  were  com- 
plete. Often,  in  fact,  this  degree  of  freedom  of  motion  is  de- 
sirable. 

Again,  consider  the  familiar  fly-ball  (conical-pendulum)  gov- 
ernor (Fig.  9)  as  used  on  many  classes  of  steam-engines.  The 
balls  of  the  governor  are  constrained  to  the  extent  that  their  cen- 
tres always  lie  in  the  surface  of  a  certain  sphere,  that  is,  they  have 
spherical  motion ;  but  they  may  move  in  any  path  whatever  (within 
the  limits  of  action),  lying  in  the  prescribed  spheres.  The  path 
in  which  they  actually  travel  depends  upon  the  relations  between 
their  mass,  angular  velocity,  and  radius  relative  to  the  axis 
about  which  they  revolve  at  the  instant  under  consideration,  and 
their  motion  can  be  determined  only  in  connection  with  these 
forces. 

In  all  but  a  few  such  cases  as  the  latter  we  may  study  con- 
strained motion  quite  apart  from  the  forces  involved  in  the  opera- 
tion of  the  mechanism. 

25.  Machine  Design;  Kinematics. — The  design  of  a  new  ma- 
chine or  the  analysis  of  an  existing  machine  divides  itself  natu- 
rally into  two  quite  distinct  processes,  as  will  appear  upon  brief 
reflection.  In  every  machine  energy  is  supplied  from  some  source 
and  so  modified  as  to  produce  some  useful  effect.  A  train  of 


CONCEPTIONS  OF  MOTION.    NATURE  OF  A  MACHINE.     35 

mechanism  is  employed  to  secure  the  required  transfer  or  modifi- 
cation, and  this  intermediate  mechanism  must  be  adapted,  first,  to 
secure  the  motion  demanded  to  produce  the  desired  result  ;  and 
second,  to  transmit  the  necessary  force  without  breakage  or  undue 
distortion  of  the  members  of  the  machine.  It  will  readily  appear 
that  the  motion  system  can  often  be  planned  or  studied  without 
^considering  the  magnitude  of  the  forces  transmitted.  As  an  illus- 
tration, consider  the  lever  of  Fig.  35,  in  which  the  distance  from 

the  fulcrum,  0,  to  A  is,  say  1  foot, 
and  distance  from  0  to  B  is  2  feet. 
Now  if  B  be  moved  through  a  dis- 
tance  of  2",  A  will  move  through 
1".  Suppose  the  resistance  to  act 
at  A,  and  the  force  which  over- 
comes it  at  B.  The  resistance  at  A 

Fig.  35 

may  be  1  pound,  1  ton,  or  of  any 

-other  amount  whatsoever,  then  the  force  required  at  B  to  overcome 
it  will  be  of  a  corresponding  magnitude;  but  in  any  case  the  ratio 
of  the  motions  or  the  velocity  ratio  of  A  to  B  will  be  determined  by 
the  length  of  the  lever-arms,  independent  of  the  actual  forces  in- 
volved. Furthermore,  if  B  moves  2"  in  one  second,  A  will  move 
1  inch  in  one  second;  if  B  moves  6"  in  one  second,  A  will  move 
3"  in  one  second ;  or  if  B  makes  4  strokes  per  second,  A  will  also 
make  4  strokes  per  second;  but  the  (total)  path  described  by  A,  or 
distance  moved  through,  will  be  but  half  the  path  of  B.  So  for 
any  motion  whatever  of  B,  A  will  have  a  definite,  corresponding 
motion,  and  the  ratio  of  the  motions  will  remain  invariably  the 
s.ime,  whatever  the  actual  motion  and  the  forces  acting  may  be.  It 
appears  then  that  the  ratio  of  the  motions  which  A  and  B  have, 
relative  to  the  fixed  member,  is  determined  purely  by  geometrical 
considerations,  and  may  be  studied  without  taking  into  account 
anything  else. 

The  lever  must  not  only  give  a  required  motion  to  one  point 
for  a  given  motion  of  another,  but  it  must  transmit  a  certain  force; 
.and  having  satisfied  the  motion  requirement,  it  is  necessary  to  give 
the  lever,  the  pivot,  and  all  parts  subjected  to  load,  sufficient 
strength  to  safely  carry  the  loads.  This  second  operation  requires 
&  knowledge  and  application  of  the  physical  properties  of  the  mate- 


36  KINEMATICS  OF  MACHINERY. 

rials  used,  and  of  other  laws  of  mechanics  than  those  relating  to 
simple  motion. 

A  similar  discussion  would  apply  to  all  ordinary  mechanisms, 
but  the  foregoing  is  perhaps  sufficient  for  the  present  purpose, 
viz.,  showing  how  the  design  of  a  machine  may  be  taken  up  as 
two  distinct  processes,  one  of  which  can  be  completed,  subject  to 
certain  modifications,  before  the  other  is  considered. 

Frequently  the  actual  motions,  velocities,  and  accelerations,  as 
well  as  the  external  loading,  are  involved  in  the  second  process; 
for  the  weight  of  a  part  and  the  changes  in  direction  and 
velocity  of  its  motion  produce  stress  in  the  restraining  members. 
An  example  is  the  stress  due  to  "centrifugal  force."  In  the 
complete  design  of  a  machine,  there  are  many  other  considerations 
affecting  the  durability,  freedom  from  frictional  and  other  losses, 
etc.,  that  are  no  less  important  than  the  preceding,  and  all  of  these 
must  be  carefully  weighed  in  their  proper  place;  but  these  consid- 
erations are  not  within  the  province  of  the  present  work. 

The  two  grand  divisions  of  the  Mechanics  of  Machinery  out- 
lined above  are  called: 

(I)  The  Geometry  of  Machinery,  Pure  Mechanism,  or  Kine- 
matics ; 

(II)  Constructive  Mechanism,  or  Machine  Design. 

In  beginning  the  study  of  machinery,  it  is  both  logical  and 
convenient  to  take  up  the  above  divisions,  in  the  order  given. 
The  first  division,  Geometry  of  Machinery,  Kinematics,  or  Machine 
Motions,  will  be  the  leading  subject  of  the  present  work. 

While,  as  in  the  illustration  of  the  lever  given  above,  the  con- 
sideration of  the  forces  acting  is  not,  generally,  involved  in  the 
study  of  machine  motions,  there  are  important  classes  of  mechan- 
isms the  motions  of  which  cannot  be  treated  without  taking  cog- 
nizance of  certain  forces.  Examples  of  these  are  the  centrifugal 
governors,  already  mentioned,  so  commonly  used  on  steam-engines 
and  other  motors,  in  which  centrifugal  force  and  the  force  exerted 
by  springs,  or  by  gravity,  can  not  be  separated  from  a  treatment  of 
the  motions;  also  escapements  such  as  are  used  in  clocks  and 
watches. 


CHAPTER   II. 
GENERAL  METHODS    OF  TRANSMITTING  MOTION  IN  MACHINE?. 

26.  Transmission    through    Space   without   Material   Connec- 
tion.— In  most  mechanisms  motion  is  transmitted  and  controlled 
through  actual  contact  of  members  of  the  mechanism;  but  there 
are  certain  exceptions  as,   for  example:  electric  motors,   escape- 
ments of  clocks,  governors,  etc.     The  armature  of  the  motor  is 
caused   to   rotate  by   electromagnetic   forces    acting   across    an 
open    space;    the    pendulum    or    balance-wheel    of    the    clock- 
work is  driven  by  the  intermittent  action  of  gravity  or  a  spring, 
though  the  resultant  motion  is  affected  by  the  length  of  the  pen- 
dulum or  proportions  of  the  wheel,  independently  of  the  intermit- 
tent connection  with  the  escape- wheel ;    and  the  motion  of  the 
governor-balls  is  determined  by  the  combined  effect  of  centrifugal 
force  and  of  gravity  or  of  springs.     In  these  instances,  the  motion 
of  the  members  is  not  fully  constrained,  and  the  motions  of  such 
mechanisms  cannot  be  treated  by  purely  geometric  methods,  inde- 
pendently of  the  forces  involved  in  the  actual  operation.     With 
the  exceptions  of  these  and  similar  mechanisms,  motion  is  only 
transmitted  by  direct  contact  of  one  material  body  with  another. 
We  are  at  present  only  concerned  with  such  cases  as  the  latter. 

27.  Transmission  by   Actual   Contact. — These    cases    may  be 
conveniently  treated  under  two  divisions;  and  the  second  division 
may  be  subdivided  into  two  classes. 

In  every  mechanism  we  have  one  member, — frequently  called 
a  link,  whatever  its  form — driving  another  link;  the  former  is 
called  the  driver,  and  the  latter  the  follower. 

The  driver  may  have  a  surface  which  bears  directly  upon  the 
follower,  or  there  may  be  an  intermediate  piece  serving  to  transmit 
the  force  and  motion.  This  intermediate  connector  may  be  a  rigid 

37 


38 


KINEMATICS  OF  MACHINERY. 


bar  or  block ;  it  may  be  a  flexible  band  (as  a  belt,  cord,  or  chain) ; 
or  it  may  be  a  confined  fluid  column. 

These  various  modes  of  connection  give  rise  to  the  following 
classification : 

Direct  Contact.         .  * 


Modes  of 

Transmission 

of  Motion. 


Intermediate 
Connectors 


"  Rigid  Links. 


Flexible          |  Bands'  etc' 

Connectors 


Fluid  Oonnectors. 
28.  Higher  and  Lower  Pairing. — Figs.  36  and  37  represent 


Fig.  36 


Fig.  37 


examples  of  direct  contact  transmission,  in  which  A  will  be  con- 
sidered as  the  driver  and  B  as  the  follower.  The  contact  in  these 
examples  is  confined  to  a  line  (or  a  point),  instead  of  being  dis- 
tributed over  a  finite  surface. .  Such  contact — line  or  point  con- 
tact— constitutes  what  is  termed  higher  pairing ;  while  contact 
over  a  finite  surface  is  called  lower  pairing.  Higher  pairing  usu- 
ally involves  greater  wear  at  the  contact  surfaces,  and  is  generally 
to  be  avoided  if  it  is  possible  to  do  so.  The  contact  between 
the  teeth  of  gears  and  that  between  most  cams  and  their  fol- 
lowers is  necessarily  higher  pairing.  However,  there  are  many 
cases  where  it  is  perfectly  practicable  to  introduce  modifications  in 
the  construction  which  distribute  the  contact  over  a  surface,  with- 
out sacrifice  of  the  kinematic  relations.  While  this  does  not  change 
the  relative  motion  of  driver  and  follower,  it  is  practically  advan- 
tageous in  reducing  the  intensity  of  contact  pressure,  and  conse- 
quently the  wear  of  parts.  It  is  usually  desirable  to  substitute 
lower  pairs  for  higher  pairs  where  practicable.  In  certain  cases, 


TRANSMITTING  MOTION  IN  MACHINES 


39 


where  pure  surface  contact  is  not  possible,  a  modification,  which 
does  not  eliminate  line  contact,  may  be  advantageously  employed. 

Figs.  38  and  39  show  cases  of  transmission  from  the  driver  A 
to  follower  B  by  direct  contact,  higher  pairing  being  used.     In 


Fig.  38 


Fig.  39 


these  cases  (Figs.  38  and  39),  the  kinematic  action  is  the  same  as 
would  result  from  contact  between  the  point  p  of  driver  and  the 
dotted  "pitch-line"  of  the  follower,  as  indicated  by  Figs.  40  and 


Fig.  40 


Fig.  41 


41.  These  latter  figures  do  not  represent  practical  mechanisms, 
for  of  course  it  is  necessary  to  have  the  contact  parts  of  sen- 
sible size. 

Figs.  42  and  43  show  similar  arrangements,  each  having  a  suit- 
able block  interposed  between  the  driver  and  follower.  These  in- 
termediate pieces  do  not  change  the  transmission  of  motion  in  any 
degree,  but  it  will  be  noticed  that  the  driver  now  acts  upon  the 
block,  and  the  block  upon  the  follower,  eliminating  line  contact 
entirely  without  sacrifice  of  the  desired  motion,  and  a  better  prac- 
tical mechanism  is  thus  obtained.  The  mechanisms  of  Figs.  38  and 
39  are  respectively  identical,  kinematically,  with  those  of  Figs.  42 
and  43.  An  intermediate  connector,  or  a  new  link,  has  been 


KINEMATICS  OF  MACHINERY. 


introduced,  and,  in  a  sense,  the  mechanism  comes  under  the  second 
division  in  the  above  classification;  but  this  intermediate  con- 
nector, C,  does  not  alter  the  transmission  of  motion.  As  we  are 
not  concerned  in  the  least  with  the  motion  of  this  block  itself,  it 


Fig.  42 


Fig.  43       ^-_| -' 


may  be  neglected  in  the  kinematic  analysis,  and  such  substituted 
mechanisms  will  be  treated  as  direct  contact  under  Division  I.  If 
desired,  A  could  be  treated  as  the  driver  of  (7;  and  C  (which  is  the 
follower  with  regard  to  A)  as  the  driver  of  B. 

Except  in  cases  where  the  contact  surfaces  of  both  links  are 
either  planes,  surfaces  of  revolution,  or  regular  helical  surfaces, 
such  substitution  cannot  be  made;  for  these  are  the  only  contact 
surfaces  possible  in  lower  pairing.  In  gear-teeth,  for  example,  it 
is  not  possible  to  avoid  higher  pairing. 

There  are  other  cases,  as  in  cams,  where  it  is  practically  of 
advantage — even  though  line  contact  is  not  thus  eliminated — 
to  introduce  an  intermediate  piece,  replac- 
ing one  kind  of  line  contact  by  another 
kind.  Thus,  in  Fig.  44,  the  cam  could  act 
directly  upon  the  end  of  the  rod  B;  but  the 
friction  would  be  excessive  and  the  action 
would  not  be  smooth,  especially  if  the  form 
of  the  cam  departs  much  from  that  ,of  a 
surface  of  revolution  whose  geometrical  axis 
coincides  with  the  axis  of  rotation.  When 
a  roller,  C,  of  suitable  size,  is  attached  to 
Fi9-  44  the  end  of  the  rod  much  smoother 

action  is  obtained.     The  roll  does  not  rub  on  the  cam,  as  in  the 


TRANSMITTING  MOTION  IN  MACHINES. 


41 


direct  sliding  contact  of  the  follower  upon  the  driver,  and  the 
sliding  action  is  transferred  to  the  pin  which  carries  the  roll,  where 
surface  contact  is  procured.  In  this  case,  as  in  those  of  Figs.  42 
and  43,  the  intermediate  connector  can  be  neglected  kinematically. 
The  motion  transmitted  to  the  follower  corresponds  to  the  contact 
of  the  centre  of  the  pin,  p,  with  a  hypothetical  surface — called  the 
pitch  surface  of  the  cam — indicated  by  the  dotted  line.  The  rela- 
tion of  this  pitch  surface  to  the  actual  working  surface,  and  the 
derivation  of  the  latter  from  the  former,  will  be  treated  later  under 
the  head  of  Cams. 

In  the  following  discussions  of  the  angular  velocity  ratio  of 
driver  to  follower,  in  direct-contact  mechanisms,  the  auxiliary  con- 
nector— the  block,  cam-roll,  etc. — will  be  neglected,  as  it  has  been 
seen  that  its  own  motion  is  immaterial,  and  that  it  does  not  affect 
the  velocity  ratio  of  driver  to  follower. 

It  is  interesting,  in  connection  with  the  preceding  discussion,  to 
note  that  it  is  sometimes  advantageous  to  employ  line  or  point 
contact  when  the  case  will,  kinematically,  permit  surface  contact. 
The  familiar  roller-bearings  and  ball-bearings  are  examples.  In 
these,  friction  and  wear  are  reduced  by  the  substitution  of  line  or 
point  contact  for  the  ordinary  surface-bearing,  because  by  this  sub- 
stitution the  grinding  effect  of  sliding  is  replaced  by  a  rolling  of 
each  member  upon  those  with  which  it  comes  into  contact. 

29.  Direct-contact  Transmission. — The  most  general  case  of 
direct  contact  is  between  two  surfaces  such  as  are  shown  in  Figs, 


Fig.  45 


Fig.  46 


36,  37,  and  45.  The  surfaces  may  both  be  convex,  or  one  may  be 
concave,  as  in  Fig.  45 ;  but  in  the  latter  event  the  radius  of  curva- 
ture of  the  concave  surface  must  always  be  at  least  as  great  as  that 
of  any  portion  of  the  other  member  that  can  come  in  contact  with 


42 


KINEMATICS   OF  MACHINERY. 


it ;  otherwise  a  certain  part  of  the  concave  surface  will  not  come 
into  contact,  as  in  Fig.  46,  between  e  and  /,  and  the  action  will 
be  discontinuous  or  irregular.  Except  for  this  limitation,  the 
surfaces  may  be  of  any  form;  but  the  present  discussion  will  be 
confined  to  those  cases  in  which  all  the  elements  of  the  contact 
surfaces  and  both  axes  of  rotation  are  parallel  to  each  other. 
Members  having  such  contact  surfaces  can  have  only  plane 
motion  relative  to  each  other.  There  are  special  cases  not  coming 
under  this  class  which  will  be  treated  later  in  the  work. 

In  treating  these  plane  motions  the  simplification  referred  to 
in  Art.  10  can  be  applied,  that  is,  the  representation  of  these 
surfaces  and  their  motions  by  their  projection  on  a  plane  parallel 
to  the  plane  of  motion. 

Motion  can  be  transmitted  by  direct  contact  only  by  normal 
pressure  between  the  surfaces.  The  action  between  the  two 
parts  in  contact  may  have  the  nature  of  rolling,  sliding,  or  mixed 
rolling  and  sliding.  The  last  condition  is  the  most  general.  The 
precise  nature  of  these  actions  and  the  method  of  determining 
them  will  form  the  subject  of  a  later  section. 

Referring  to  Fig.  47,  it  is  evident  that  all  points  in  A  must 
rotate  about  0,  and,  likewise,  all  points  in  B  must  rotate  about  0' '. 
Consequently  the  velocity  of  any  point  in  either  is  represented  by 

a  line  through  that  point  perpen- 
dicular to  the  radius  connecting 
it  with  its  centre,  0  or  0',  as  the 
case  may  be. 

The  point  of  contact  between 
the  two  members  is  at  P,  which 
may  be  regarded  as  the  coin- 
cident position  of  two  points, 
one  of  which,  called  Pa,  is  a 
point  in  A,  while  the  other, 
called  Pb,  is  a  point  in  B.  Then  Pm  and  Pn  represent  the  veloc- 
ities of  Pa  and  Pb,  respectively. 

The  pressure  between  A  and  B  is  transmitted  in  the  direction  of 
the  common  normal  to  the  two  surfaces  at  the  point  of  contact,  and 
whatever  the  actual  velocities  of  Pa  and  P&,  the  components  of 


Fig.  47 


TRANSMITTING   MOTION  IN    MACHINES. 


43 


these  two  velocities  along  the  line  of  this  normal  (NNf)  must  be  equal 
when  they  are  in  contact  (as  Ps).  If  the  normal  component  of 
the  velocity  of  Pb  were  greater  than  the  normal  component  of  the 
velocity  of  Pa,  B  would  quit  contact  with  A .  On  the  other  hand, 
if  the  normal  component  of  the  velocity  of  Pa  is  greater  than  that 
of  Pb,  A  would  enter  the  space  occupied  by  B,  and  this  is  incon- 
sistent with  our  conception  of  a  rigid  body. 

As  we  are  concerned  only  with  the  velocity  ratio  of  the  two 
members,,  and  as  this  ratio  is  independent  of  the  actual  velocities, 
the  velocity — either  angular  or  linear — of  one  member  may  be 
assumed  if  not  known,  and  this  affords  a  means  of  determining  the 
velocity  of  the  other  member  at  that  instant.  The  angular  velocity 
ratio  of  the  driver  to  the  follower  is,  in  the  general  case,  varying 
continually.  Simple  methods  of  determining  this  angular  velocity 
ratio  at  any  phase  of  the  motion  may  be  used,  and  a  close  analogy 
exists  in  these  methods  as  applied  to  the  three  different  classes  of 
transmission.  Each  class  will  be  discussed  by  itself,  and  the  gen- 
eral relation  will  be  deduced  afterward. 


T' 


Fig.  48 


Fig.  49 


If  in  Figs.  48  and  49,  oo^  =  the  angular  velocity  of  A,  be 
known,  for  the  phase  under  consideration,  the  linear  velocity  of 
Pa  can  be  found  from  the  relation : 


linear  vel. 

ang.  vel.  =  =-. : 

radius. 


or 


Pm 
OP' 


Represent  the  linear  velocity  of  Pa  by  Pw,  and  resolve  it  into  its 
components  along  and  perpendicular  to  the  common  normal 


44  KINEMATICS   OF  MACHINERY. 

Thus  the  normal  component  Ps,  and  the  tangential  component  Pta 
are  obtained.  The  direction  of  the  motion  of  P&  is  known  (perpen- 
dicular toPO'),  and  the  normal  component  of  its  velocity  must  equal 
that  of  Pa,  or  it  is  Ps,  hence  the  actual  velocity  of  P&,  Pn,  can  be 
found  (Art.  17,  Case  («));  and  its  tangential  component,  Ptbf  may 
be  found  from  this  if  desired.  Having  found  in  this  way  the 
linear  velocity  of  P6,  its  angular  velocity,  &>2,  may  be  obtained  by 
dividing  this  quantity  by  the  radius  PO'  ;  and  the  angular  velocity 

ratio  of  A  to  B,  for  this  phase  of  the  motion  =  —  %  becomes  known. 

fi», 

A  similar  method  could  be  employed  in  determining  this  ratio  for 
any  number  of  phases,  and  thus  the  motion  of  the  follower,  cor- 
responding to  the  motion  of  the  driver,  whether  uniform  or  other- 
wise, could  be  derived.  The  following  demonstration  establishes 
relations  of  the  angular  velocity  ratio  of  driver  and  follower,  in 
direct-contact  mechanisms,  which  are  much  more  expeditious  and 
convenient  in  drawing-board  practice. 

In  Figs.  48  and  49,  let  Pm  and  Pn  represent  the  linear  veloci- 
ties of  P0  and  P6,  respectively.  Drop  perpendiculars  Of  and  O'g 
from  0  and  0'  upon  the  normal  NN'.  Ps  is  the  common  normal 
component  of  Pm  and  Pn.  Pms  and  6/P/are  similar  triangles; 
also  Pns  and  O'Pg  are  similar  triangles. 

col  =  angular  velocity  of  Pa  about  0  =  -       =  --.,   .     (1) 


a  =  angular  velocity  of  B  about  0'  =  =  --.  .     (2) 


- 

Of       Ps  ~  -  Of 


Prolong  the  normal  and  line  of  centres  till  they  intersect  at  /; 
then  lOf  and  10'  g  are  similar  triangles  and 


-  =        =.  (4) 

10        Of      co, 


TRANSMITTING  MOTION  IN   MACHINES. 


45 


It  follows  from  the  above  discussion  that:  In  direct-contact 
mechanisms  the  angular  velocities  of  the  members  are,  at  any  phase, 
inversely  as  the  perpendiculars  let  fall  from  their  fixed  centres  upcn 
the  line  of  the  common  normal  of  the  two  curves  ;  or  inversely  as  the 
segments  into  which  the  line  of  centres  is  divided  ~by  this  normal. 

>  30.  Link-connectors. — A  relation  very  similar  to  that  just  de- 
rived can  be  deduced  for  the  angular  velocities  of  a  driver  and  fol- 
lower connected  by  a  rigid  link. 
In  this  case  we  are  not  concerned 
with  the  motion  of  the  inter- 
mediate connector  itself. 

Figs.  50  and  51  show  two  arms, 
OA  and  O'B,  free  to  turn  about 
the  fixed  centres  0  and  Or,  and 
connected  by  the  link  AB.  The 
velocity  of  the  point  A  is  rep-  F|9- 50 

resented  by  Am,  perpendicular  to  OA.     The  velocity  of  B  is  shown 
by  Bn,  perpendicular  to  O'B',  and  its  magnitude  is  determined  by 


the  fact  that  its  component  in  the  direction  of  AB  must  equal  the 
component  of  Am  in  this  same  line ;  for  if  these  components  are  not 
equal,  the  distance  between  A  and  B  must  change,  which  is  incon- 
sistent with  the  conception  of  a  rigid  body;  hence,  if  Am  is 
assumed,  En  is  thereby  determined. 


46  KINEMATICS  OF  MACHINERY. 

Let  G?,  =  angular  velocity  of  A  about  0  =  -j,  .     .     (1) 


Tt 

Let  a?,  =  angular  velocity  of  B  about  0'  =  --=.      .     (2) 


Drop  perpendiculars  0/and  O'g  upon  AB-,  then  triangles  OAf 
and  .4ms  are  similar;  also  0'  Bg  and  Bnr  are  similar. 

From  #4/"  and  Ams,  ~      =  -=GJi»    •     •     •     (3) 


From  O'Bg  and  Bnr=          =  ^    ...     (4) 


Produce  -4.5  and  00'  (if  necessary)  to  intersect  in  /;  then 

10'  __  0^  _  «, 

/O    "    0/  << 

From  this  reasoning  the  following  statement  is  drawn  : 

In  two  arms  connected  ~by  an  intermediate  link  the  angular  ve- 
locities of  the  arms  are  to  each  other  inversely  as  the  perpendiculars 
let  fall  from  the  fixed  centres  upon  the  line  of  the  link  ;  or  inversely 
as  the  segments  into  which  the  line  of  centres  is  cut  by  the  line  of  the 
link  (both  of  these  lines  produced,  if  necessary). 

These  relations  may  be  seen  from  direct  inspection,  by  assuming 
the  system  to  be  replaced  by  the  two  effective  arms,  Of  and  O'g, 
connected  by  the  link  fg.  This  new  system  would  evidently  be 
equivalent  to  the  original  system,  for  this  particular  position  (but 
for  no  other)  ;  and  as  the  arms  Of  and  O'g  are  perpendicular  to  the 
link,  the  linear  velocities  of  /  and  g  are  equal  ;  hence  the  angular 

velocities  of  the  arms  are  inversely  as  the  radii,  or  —  •  =  -^.     This 

would  apply  to  any  phase;  but  the  substituted  arms,  Of  and  O'g, 
would  not  be  of  the  same  lengths  for  different  phases. 


TRANSMITTING  MOTION  IN  MACHINES.  47 

The  relation  deduced  above  may  be  arrived  at  also  by  means  of 
the  method  of  instantaneous  axes.     In  Figs.  52  and  53,  co,  and  a?t 


have  the  same  signification  as  before,  and  co  =  angular  velocity  of 
the  connecting-link  about  its  instant  centre. 

A  and  B  are  two  points  in  the  connector  AB  ;  therefore  the 
motions  of  these  two  points  completely  determine  the  motion  of 
this  link.  The  velocity  of  A  is  Am,  perpendicular  to  OA ;  and  the 
velocity  of  B  is  Bn,  perpendicular  to  O'B.  Therefore  Q,  at  the 
intersection  of  OA  and  0' B,  is  the  instant  centre  for  the  link  AB 
in  the  phase  shown  (see  Art.  19).  As  the  angular  velocity  equals 
the  linear  velocity  divided  by  the  radius : 


Am       Bn 


GO  =  -7r-r  = 


QA 


(7) 


Drop  perpendiculars  Qlc,  Of,  and  O'g  from  Q,  0,  and  #', 
respectively,  upon  the  line  of  the  link  AB. 

OAf  and  QAk  are  similar  triangles;  also  O'Bg  and  QBTc  are 
similar. 


.     «i      4m      QA_QA_Qk 

ca    '  OA      Am  ~  OA  ~  Of 


.     .      (8) 


(*>__    Bn_  __  _ 

5;  ==  QB  X  ~Bn  ~  ~QB  "  Qk' 


KINEMATICS  OF  MACHINERY. 


From  equations  (8)  and  (9), 


=  ^  =  77T-    •     •     •      (W) 


CO 


This  last  demonstration  thus  gives  a  result  identical  with  equa- 
tion (6). 

^31.  Wrapping-connectors.  —  This  term  includes  belts,  bands, 
ropes,  chains,  and  all  flexible  members  used  to  connect  a  driver  and 
follower,  and  transmitting  force  only  under  tension.  The  working 
surfaces  may  be  of  any  convex  cylindrical  form  ;  but  concave  forms 
are  excluded,  as  the  wrapper  would  not  follow  the  depressions  of 
such  a  form,  and  if  used  the  action  would  not  be  smooth  and  con- 
tinuous. 

The  term  cylindrical  as  used  above  applies  strictly  in  case  of 
flat  bands.  In  case  of  round  cords,  ropes,  etc.,  the  contact  surface 

is  usually  grooved  to  correspond 
x  more  or  less  closely  to  the  form  of 
the  wrapper,  but  the  motion  is  in 
this  case  equivalent  to  that  which 
would  be  obtained  by  the  neutral 
axis  of  the  connector,  wrapping 
upon  an  ideal  pitch  line  of  the 
member  upon  which  it  is  carried 
(see  Fig.  54).  The  mathematical  (and  kinematic)  wrapper,  or  the 
pitch  line,  is  the  line  xxx. 

In  case  of  flat  bands,  also,  the  true  pitch  surface,  and  line  of 
action,  are  at  a  distance  from  the  physical  face  of  the  rigid  mem- 
ber or  carrier,  equal  to  about  one-half  the  thickness  of  the  band. 
For  the  present  purpose  the  connector  will  be  treated  as  of  no 
sensible  thickness,  and  the  surfaces  shown  in  Figs.  55  to  58  are  to- 
be  taken  as  the  true  pitch  surface.  The  effect  of  thickness  of  con- 
nector will  be  discussed  in  a  later  chapter. 

The  band  is  flexible,  but  is  supposed  to  be  practically  inexten- 
sible;  and  as  it  is  subjected  only  to  tension,  the  distance  between 
any  two  points  of  the  band  cannot  change.  This  implies  that, 
whatever  actual  velocities  two  such  points  may  have,  the  components- 


-  54 


TRANSMITTING  MOTION  IN  MACHINES.  49 

of  these  velocities  in  the  direction  of  the  connector  must  be  the  same 
at  any  instant. 

In  Figs.  55  and  56,  a  is  the  driver  and  I  is  the  follower.  Either 


Fig.  55 


Fig.  56 


of  the  tangent  points  of  the  band  and  carrier  (A  or  B)  is  a  coinci- 
dent point  of  the  band  and  of  the  member  which  it  meets  at  that 
point;  GL>J  and  &?.,  are  the  angular  velocities  of  the  points  A  and  B 
respectively,  and  Am  and  Bn  are  the  corresponding  linear  veloc- 
ities. 

Am  Bn 


Since  A  and  B  are  two  points  in  the  inextensible  band,  their 
components  of  velocity  in  the  direction  of  the  connector  are  equal, 
or  As  =  Br.  Drop  perpendiculars  from  0  and  0'  upon  the  line  of 
the  connector ;  then  OAf  and  Ams  are  similar  triangles ;  also  0'  Bg 
and  Bnr  are  similar. 

col  =  angular  velocity  of  A  about  0  =  7  -.  =  -j^,    .     (1) 

(JA        Uj 

?s  =  angular  velocity  of  B  about  0'  =  -       =     r*,   .     (2) 


oo,  ~0f 


*/ 

w 


=  £r). 


(3) 


50  KINEMATICS  OF  MACHINERY. 

Prolong  the  line  of  centres,  00',  and  the  line  of  the  band,  ABt 
to  meet  in  / ;  then  lOf  and  10 'g  are  similar  triangles. 

10'  _0'g    _  a?, 
•''7o^~~0f~^ *' 


From  equations  (3)  and  (4)  we  can  formulate  the  statement: 
In  wrapping  -connectors,  the  angular  velocity  of  the  driver  is  to 
that  of  the  follower  inversely  as  the  perpendiculars  let  fall  from  the 
fixed  centres  upon  the  line  of  the  connector  ;  or  inversely  as  the  seg- 
ments into  which  the  line  of  centres  is  cut  by  the  line  of  the  connector 
(both  produced  if  necessary). 

This  relation  may  be  shown  by  the  instantaneous  centre  method 
also  (Figs.  55  and  56).  A,  as  a  point  in  the  driver,  has  a  linear 
velocity  Am  and  GOI  =  Am  -f-  OA.  B,  as  a  point  in  the  follower, 
has  a  linear  velocity  Bn,  and  o?3  =  Bn  -f-  O'B.  The  acting  part 
of  the  connector,  AB,  has  an  angular  velocity  about  Q  of 

Am       Bn 
=  QA  '-=  OB' 

Let  fall  Qk  perpendicular  to  AB  ;  then  OAfand.  QAk  are  similar; 
also  O'Bg  and  QBk  are  similar. 

«,  _  Am        QA       QA_Qk 

GO        OA    x  Am  ~  OA  ~  Of9  {  } 

GO        Bn        O'B      O'B      O'g 

_  —  _  v  _  —  -  —  —  _  —  i  fi  i 

GO,  ~  QB  '     Bn        QB~  Qk' 


Multiply  (5)  by  (6)  : 


C*  v  O'g  __  O'g  _  10' 
Of  X  Qk  -  Of  '-  10' 


This  result  accords  with  that  of  equation  (4). 

32.  Similarity  of  Expressions  for  Angular  Velocity  Eatio  in  the 
Three  Modes  of  Transmission.  —  By  substituting  the  term  line  of 
action  for  "line  of  the  normal,"  "line  of  the  link,"  and  "line  of 
wrapping-connector,"  in  the  three  cases  of  direct  contact,  link-con- 


TRANSMITTING  MOTION  IN  MACHINES. 


51 


nector,  and  wrapping-connector,  respectively;  the  following  state- 
ment will  apply  to  all  of  these  modes  of  transmitting  motion  : 

The  angular  velocities  of  the  members  are  inversely  as  the  per- 
pendiculars  let  fall  from  their  fixed  centres  upon  the  line  of  action  ; 
or  inversely  as  the  segments  into  which  the  line  of  action  cuts  the 
line  of  centres. 

There  are  special  cases  in  which  the  preceding  theorems  are  not 
available,  because  the  expressions  become  indeterminate;  though 
these  cases  can  be  reconciled  to  the  general  statement.  For  ex- 
ample :  see  the  direct-contact  mechanism  of  Fig.  57,  or  the  link- 
work  of  Fig.  58.  In  these  mechanisms  the  centre  about  which  B 


Fig,  57 


Fig.  58 


rotates  is  at  QO  ,  hence  the  perpendicular  from  it  upon  the  normal 
=  QO  (also  the  segment  from  its  centre  to  the  line  of  action  =  00); 


and  by  the  theorem  of  Art.  29;  —  •  =  n 


=  o> 


This  is  consistent, 


as  the  angular  velocity  of  the  follower  B,  is  o;  hence  that  of  the 
driver  (A)  is  infinitely  greater  than  that  of  the  follower;  but  the 
result  does  not  enable  us  to  derive  the  linear  velocity  of  the  follower, 
for  this  equals  the  angular  velocity  multiplied  by  the  radius,  =  o 
X  oo  ,  an  indeterminate  expression.  The  linear  velocity  of  the  fol- 
lower is  easily  found  by  other  means,  however,  as  its  normal  com- 
ponent must  equal  that  of  the  linear  velocity  of  the  driver,  and  the 
direction  of  the  follower's  motion  is  known.  From  this  data,  the 
linear  velocity  of  the  follower  is  derived  (see  Art.  17).  A  similar 
course  of  reasoning  applies  to  the  mechanism  shown  in  Fig.  58. 


52  KINEMATICS  OF  MACHINERY. 

In  the  linkwork  shown  by  Fig.  59  the  follower  is  not  under 
control  of  the  driver  at  the  particular  phase  there  represented.     As 


Fig.  60 


A  reaches  the  position  shown  (at  either  dead  centre),  it  has  no 
component  of  motion  in  the  line  of  the  link  AB,  hence  there  is  no 
component  compelling  motion  of  B.  As  A  passes  this  position,  B 
might  be  moved  in  either  of  the  two  directions  indicated  by  Bn  or 
Bri.  If  the  shaft  about  which  B  rotates  is  provided  with  a  fly- 
wheel, or  similar  device,  its  direction  of  motion  may  be  maintained, 
and  as  soon  as  the  dead  centre  is  passed,  A  again  exerts  an  influence 
over  its  motion.  In  the  case  of  Fig.  60  B  comes  to  rest  as  A  passes 
the  dead  centre,  Bn  being  zero  at  this  phase. 

33.  Directional  Relation. — It  will  be  seen  by  reference  to  Figs. 
48,  50,  and  55  that  the  driver  and  follower  both  rotate  in  the  same 
direction;  that  is,  loth  members  are  turning  in  the  right-handed, 
clockwise,  or  negative  direction  as  angles  are  usually  reckoned  ;  or 
both  rotate  in  the  reverse  direction.*  It  will  also  be  noticed 
that  in  the  cases  shown  in  Figs.  48,  50,  and  55,  the  two  fixed  cen- 
tres lie  on  the  same  side  of  the  line  of  action. 

In  Figs.  49,  51,  and  56,  on  the  other  hand,  the  fixed  centres 
lie  on  opposite  sides  of  the  line  of  action,  and  the  two  members  ro- 
tate in  opposite  directions;  if  one  member  has  right-handed  rota- 
tion, the  other  has  left-handed  rotation,  and  vice  versa.  From  this 
observation  of  all  of  these  general  cases,  the  following  statement  in 

*  The  follower  is  converted  into  the  driver  when  such  reversal  takes  place 
in  Figs.  48  and  49,  but  not  necessarily  in  Fig.  50.  That  this  conversion  does 
not  take  place  in  all  direct  contact  and  wrapping-connector  mechanisms  is  evi- 
dent from  Figs.  39  and  204-209,  in  which  either  member  may  be  the  driver. 


TRANSMITTING  MOTION  IN  MACHINES.  53 

regard  to  the  Directional  Relation  is  quite  evident:  In  any  of  the 
three  ordinary  modes  of  transmission  of  motion  the  directions  of  rota- 
tion of  the  driver  and  follower  are  the  same  if  the  fixed  centres  ofboih 
lie  on  the  same  side  of  the  line  of  action  ;  and  the  directions  are 
-opposite  if  these  centres  lie  on  opposite  sides  of  the  line  of  action. 

34.  Condition  of  Constant  Angular  Velocity  Ratio. — It  has  been 
shown  that  with  any  of  the  three  common  methods  of  transmitting 
motion  the  angular  velocities  of  the  members  are  inversely  as  the 
segments  into  which  the  line  of  centres  is  cut  by  the  line  of  action. 
Thus  in  any  figure  from  48  to  56  (except  Fig.  54)  GOI  :  G0a : :  10' :  10. 

If  the  angular  velocity  ratio  is  constant,,  — -1  —  -=-^   =  a  constant, 

•and  as  00' — the  distance  between  the  fixed  centres — is  a  constant, 
/must  be  a  fixed  point  in  this  line  (or  its  extension)  in  order  that 
ihe  above  condition  be  realized.  Therefore  it  may  be  stated  that  : 
The  Condition  of  Constant  Angular  Velocity  Ratio  is  that  the  line 
•of  action  must  always  cut  the  line  of  centres  (produced  if  necessary] 
in  a  fixed  point. 

This  condition  is  fulfilled  by  an  infinite  number  of  pairs  of 
curves  which  may  be  used  as  the  outlines  of  the  acting  faces  of  di- 
rect-contact members.  It  will  appear  later  that  the  proposition  just 
•stated  is  of  fundamental  importance  in  the  theory  of  teeth  of  gears. 

The  condition  of  constant  velocity  ratio  is  fulfilled  in  the  case  of 
wrapping-connectors  when  the  driver  and  follower  have  faces  which 
are  right  cylinders  with  the  axes  of  the  cylinders  as  the  axes  of 
revolution  ;  for  example,  in  the  case  of  ordinary  pulleys  with 
crossed  or  open  belts. 

Constant  velocity  ratio  is  secured  with  link  transmission  when 
the  driving  and  the  driven  arms  are  equal  (Fig.  59),  and  the  length 
of  the  connecting-link  is  equal  to  the  distance  between  the  fixed 
centres  and  parallel  to  the  line  of  centres,  as  in  the  parallel  rods 
of  locomotives. 

35.  Nature  of  Rolling  and  Sliding. — When  two  pieces  act  to- 
gether by  direct  contact  they  may  roll  upon  each  other,  they  may 
slide  upon  each  other,  or  they  may  move  relative  to  each  other 
with  a  combined  rolling  and  sliding  action. 

Fig.  61  shows  two  such  members,  in  which  p  is  the  contact 


54  KINEMATICS  OF  MACHINERY. 

point  in  the  phase  shown.  If  r  and  s  are  any  two  points  which 
meet  as  the  action  continues  (becoming  coincident  contact  points), 
the  arcs  pr  and  ps  must  be  equal  if  the  action  is  pure  rolling.  If 


for  any  increment  of  motion  the  corresponding  arcs  of  action  of  the 
two  curves  are  not  equal,  there  must  be  some  sliding  between  them. 
In  pure  rolling  action  no  one  point  of  either  body  comes  in  contact 
with  two  successive  points  of  the  other. 

If  a  point  of  one  of  the  bodies  comes  in  contact  with  all  suc- 
cessive points  of  the  acting  surface  of  the  other  (within  the  limits 
of  its  path),  the  action  is  purely  sliding;  for  example,  the  piston 
in  the  cylinder  of  an  engine. 

In  some  cases,  as  in  many  cams  and  all  gear-teeth,  the  action 
is  mixed  sliding  and  rolling.  The  sliding  action  must  occur  along^ 
the  common  tangent  at  the  point  of  contact  of  the  two  surfaces. 

36.  Rate  of  Sliding  and  Condition  of  Pure  Rolling. — It  has- 
been  shown  that  in  direct-contact  mechanisms  the  normal  compo- 
nents of  the  velocities  of  the  points  of  contact  must  be  equal. 
The  tangential  components  may  have  any  values,  either  in  the 
same  or  in  opposite  directions.  When  the  tangential  components 
of  the  velocities  of  the  contact  points  are  in  the  same  direction  and 
equal  there  is  no  sliding,  and  the  two  velocities  are  identical  as 
corresponding  components  are  equal. 

The  rate  of  sliding  is  the  difference  of  the  tangential  components 
if  they  are  in  the  same  direction,  or  their  sum  if  they  are  in  opposite 
directions ;  or :  The  rate  of  sliding  is  the  algebraic  difference  of  the 
tangential  components  of  the  velocities  of  the  points  of  contact. 


TRANSMITTING  MOTION  IN  MACHINES. 


55 


In  direct-contact  mechanisms  the  normal  components  of  the 
velocities  of  the  points  of  contact  are  always  equal,  and  the  tangen- 
tial components  are  also  equal  when  the  action  is  pure  rolling. 
Figs.  62  and  63  illustrate  this  condition,  and  the  two  velocities,  Pm 


N" 


Fig.  63 


and  Pn,  are  identical.  But  P,  as  a  point  in  A,  moves  at  right 
angles  to  OP',  and,  as  a  point  in  B,  it  moves  at  right  angles  to 
O'P;  therefore,  when  Pm  coincides  with  Pn,  OP  and  O'P  are 
both  perpendiculars  to  the  same  line  at  the  same  point  and  they 
must  therefore  lie  in  one  right  line.  In  order  that  this  may  occur, 
P  must  lie  in  the  line  of  centres;  or:  The  condition  of  pure  roll- 
ing is  that  the  point  of  contact  shall  always  lie  in  the  line  of 
centres. 

Any  pair  of  direct-contact  pieces  bounded  by  curves  which 
satisfy  the  condition  just  stated  act  upon  each  other  with  a  pure 
rolling  action;  and  any  departure  of  the  contact  point  from  the 
line  of  centres  is  accompanied  by  sliding  action.  There  are  many 
sets  of  curves  which  may  be  employed  to  thus  transmit  motion  by 
direct  contact  and  with  pure  rolling  action,  among  which  may  be 
mentioned :  tangent  circles  (or  circular  arcs)  rotating  about  their 
centres;  pairs  of  equal  ellipses  each  rotating  about  one  of  its  foci 
with  a  distance  between  the  fixed  centres  equal  to  the  common 
major  axis;  and  pairs  of  similar  logarithmic  spirals  rotating  about 
their  foci. 

As  the  common  normal  to  two  direct-contact  members  passes 
through  the  point  of  contact,  and  as  this  point  always  lies  in  the 
line  of  centres  if  the  action  is  pure  rolling,  the  common  normal 


56  KINEMATICS  OF  MACHINERY. 

cuts  the  line  of  centres  in  the  contact  point  when  pure  rolling 
occurs.  The  angular  velocity  ratio  is  inversely  as  the  segments 
into  which  the  line  of  centres  is  divided  by  the  normal;  or  inversely 
as  the  perpendiculars  let  fall  from  the  fixed  centres  upon  the  normal 
(see  p.  45,  Art.  29).  In  pure  rolling  these  segments  are  the 
contact  radii  themselves  (OP  and  O'P  of  Figs.  62  and  63),  and 
therefore  in  such  cases  the  angular  velocities  are  inversely  as  the 
contact  radii. 

Drop  perpendiculars,  Ok  and  O'l,  from  the  fixed  centres  (Figs. 
62  and  63),  upon  the  common  tangent,  TT',  and  it  will  be  seen 

that  the  triangles  OPk  and  O'Pl  are  similar.     .'.7:7-  =-7777  ==-~  — 

OK      Or     0)2 

the  angular  velocity  ratio  of  the  members.  We  may  then  use, 
if  convenient,  the  following  relation:  The  angular  velocity  ratio, 
in  direct-contact  mechanisms  having  pure  rolling  action,  is  inversely 
as  the  perpendiculars  from  the  fixed  centres  w  the  common  tangent. 

In  the  circular-arc  forms  (Figs.  64  and  65)  the  perpendiculars 
from  the  centres  to  the  tangent  are  the  contact  radii;  thus  the 


well-known  relation  for  tangential  wheels  of  circular  section, — that 
the  angular  velocities  are  inversely  as  the  radii  of  the  circles, — is 
seen  to  agree  with  the  more  general  relations  deduced  in  this 
article. 

37.  Constant-velocity  Ratio  and  Pure  Rolling  Combined.' — 
As  stated  in  the  preceding  two  articles,  there  are  many  pairs  of 
curves  which  will  satisfy  either  the  condition  of  constant-velocity 


TRANSMITTING  MOTION  IN  MACHINES.  57 

ratio,  or  of  pure  rolling.  There  is  but  one  class  of  curves,  how- 
ever,— viz.,  circular  arcs  rotating  about  their  centres, — which  can 
have  at  the  same  time  loth  constant-velocity  ratio  and  pure  roll- 
ing (see  Figs.  64  and  65).  For  constant- velocity  ratio,  the  normal 
must  cut  the  line  of  centres  in  a  fixed  point;  for  pure  rolling,  the 
contact  point  (through  which  the  normal  passes)  must  lie  in  the 
line  of  centres.  If  both  of  these  requirements  are  met  at  the  same 
time  the  contact  point  must  be  a  fixed  point  in  the  line  of  centres; 
hence  the  contact  radii  must  be  constant;  and  therefore  the  out- 
lines of  the  members  are  circular  arcs. 

38.  Positive  Driving. — Circular-arc  members  (right  cylinders), 
as  shown  in  Figs.  64  and  65,  do  not  transmit  motion  positively. 
Actual  physical  bodies  of  the  corresponding  forms  can  transmit 
motion  from  one  to  the  other  only  through  frictional  action.  In 
the  absence  of  friction,  with  such  forms,  no  motion  could  be  trans- 
mitted against  any  resistance;  with  friction  a  limited  resistance  can 
be  overcome.  There  is  no  assurance  that  more  or  less  slipping 
may  not  occur,  and  if  this  does  take  place  the  velocity  ratio  be- 
comes both  variable  and  uncertain.* 

In  such  forms  as  those  shown  in  Figs.  62  and  63,  on  the  other 
hand,  motion  of  the  driver  involves  a  positive  and  definite  motion 
of  the  follower.  It  is  now  in  order  to  determine  the  conditions 
necessary  to  insure  positive  or  compulsory  driving.  It  is  some- 
times stated  that  positive  driving  is  only  produced  when  the  contact 
radius  of  the  driver  increases  as  the  action  proceeds  ;  thus  in  Fig. 
61  A  can  only  drive  B  positively  when  Op  is  greater  than  any  pre- 
ceding contact  radius,  as  Or,  and  less  than  any  succeeding  radius 
as  Or'.  While  this  is  the  case  with  many  forms,  it  is  not  a  general 


*  The  tangential  Component  of  the  velocity  of  either  the  driving  or  driven 
point  represents  its  rate  of  sliding  along  the  tangent.  If  the  tangential  com- 
ponents of  the  velocities  of  both  these  points  are  equal  and  in  the  same  direc- 
tion there  is  no  sliding  between  them. 

With  perfectly  smooth  surfaces,  one  of  the  members  could  not  move  the 
other  against  the  smallest  resistance.  In  the  practical  cases  where  motion  is 
transmitted  by  frictional  action,  the  effectiveness  increases  as  the  departure 
from  ideal  smooth  cylindrical  surfaces  becomes  greater.  Perfect  cylinders,  if 
such  were  possible,  would  not  be  of  the  slightest  use  in  such  causes. 


58 


KINEMATICS  OF  MACHINERY. 


requirement  for  positive  driving.  Figs.  66  and  67  show  mechan* 
isms  in  which  A  is  the  driver  and  B  is  the  follower.  It  will  be  seen 
that  A  can  rotate  indefinitely,  causing  continuous  rotation  of  B 
(though  not  with  a  uniform  velocity  ratio  between  A  and  B),  and 


Fig.  66 

at  the  end  of  each  rotation  the  two  members  will  return  to  the  same- 
relative  positions  they  had  at  the  start.  It  is  evident  then  that  the 
contact  radius  of  the  driver  cannot  increase  indefinitely.  It  will 
be  seen,  also,  that  B  may  be  the  driver,  and  that  a  similar  remark, 
will  apply  in  this  case.* 

Eeferring  to  Figs.  64,  65,  and  68,  it  is  seen  that  the  motion, 
Pm,  of  the  contact  point  of  the  driver  lies  in  the  direction  of  the 
common  tangent,  TTf  \  hence  the  normal  component  of  this  motion. 

is  zero.  It  is  only  the  normal  com- 
ponent of  the  driving  point's  motion 
which  tends  to  impart  positive  mo- 
tion to  the  follower  ;  and  in  the  cases 
of  Figs.  64,  65,  and  68,  where  the 
motion  of  this  point  is  wholly  tan- 
gential and  the  normal  component 
is  zero,  there  is  no  tendency  to  pro- 
duce positive  driving.  In  other 
words,  positive  driving  is  assured 
only  when  the  driving  contact  point  has  a  component  of  motion  in 

*  The  presence  of  the  intermediate  roll  or  block  is  not  essential,  and  the 
above  statement  would  be  equally  true  if  A  were  simply  provided  with  a  pia 
engaging  the  follower. 


Fig.  68 


TRANSMITTING  MOTION  IN  MACHINES.  59 

the  direction  of  the  normal,  and  as  the  contact  point  moves  per- 
pendicularly to  the  contact  radius,  there  can  be  no  such  normal 
component  of  motion  when  this  radius  is  perpendicular  to  the  com- 
mon tangent  ;  or,  what  is  the  same  thing,  when  this  radius  coin- 
cides with  the  common  normal.  Positive  driving  cannot  occur 
then  if  the  common  normal  passes  through  the  centre  about  which 
the  driver  rotates.  The  contact  radius  may  coincide  with  the  tan- 
gent, and  in  fact  this  is  a  very  favorable  position,  as  the  motion  of 
the  contact  point  is  then  entirely  in  the  direction  of  the  normal, 
and  there  is  no  tendency  to  slide.  If  the  common  normal  passes 
through  the  fixed  centre  about  which  the  follower  rotates  the  driver 
cannot  impart  positive  motion  to  the  follower;  for  in  this  position 
the  normal  component  of  the  motion  of  the  driven  point  is  per- 
pendicular to  the  path,  and  any  motion  in  this  direction  is  prohib- 
ited by  the  nature  of  constrained  motion.  A  force  directed  toward 
the  centre  does  not  tend  to  produce  rotation,  but  only  to  exert 
pressure  against  the  bearings. 

It  is  thus  seen  that  positive  driving  cannot  occur  if  the  common 
normal  passes  through  either  of  the  fixed  centres. 

If  the  normal  passes  through  the  centre  of  the  driver  only,  the 
driver  can  move,  but  motion  is  not  transmitted  to  the  follower. 
If  the  normal  passes  through  the  centre  of  the  follower,  the  driver 
is  locked,  for  its  motion  can  have  no  normal  component;  but  the 
follower  may  still  move  if  under  other  influences,  such  as  the  action 
of  a  fly-wheel.  If  the  common  normal  passes  through  loth  fixed 
centres,  as  in  Figs.  64  and  65,  the  motions  of  both  contact  points 
are  tangential  and  wholly  independent,  except  for  the  frictional 
action. 

In  conclusion  the  following  statement  may  be  formulated  : 
The  condition  of  positive  driving  is  that  the  common  normal  shall 
not  pass  through  the  fixed  centre  of  either  the  driver  or  the  follower. 
39.  Inversion  of  Mechanisms. — It  was  explained  in  Art.  5  that 
one  body  may  have  at  any  time  distinct  motions  relative  to  different 
bodies,  and  that  it  is  sometimes  convenient  to  refer  the  motion  of 
a  member  to  some  other  part  than  the  fixed  frame.  Take,  for 
example,  the  crank  and  connecting-rod  mechanism  of  the  ordinary 
reciprocating-engine,  as  shown  in  Figs.  27  and  69.  There  art 


60 


KINEMATICS  OF  MACHINERY. 


four  members  of  this  mechanism :  the  crank,  connecting-rod,  cross- 
head  and  piston  (the  last  two  are  kinematically  one  piece),  and  the 
frame  (including  the  cylinder).* 

In  Fig.  69  these  parts  are  designated  by  the  letters  a,  b,  c,  and 
d  respectively,  and  the  shading  of  d  is  used  to  indicate  that  it  is 

the  stationary  member.  The 
crank,  a,  rotates  about  the  centre 
Oad  relative  to  the  frame,  d,  and 
this  centre,  Oad,  is  the  instant 
centre  (also  a  permanent  centre) 
for  the  motion  of  a  relative  to  d. 
The  frame,  d,  also  rotates  rela- 
tive to  the  crank,  a,  about  this 
same  centre;  for  if  we  imagine 
the  crank  to  be  the  fixed  mem- 
ber, as  in  Fig.  70,  d  actually  does 
rotate  about  this  centre  as  the 
mechanism  operates;  but  the  change  in  the  relative  positions  of  the 
members  is  simply  that  due  to  the  mode  of  constrainment,  which- 
ever  member  is  fixed,  and  we  have  not  changed  the  form  of  the 
mechanism  in  any  way;  hence  the  relative  motions  of  the  parts  are 
the  same  under  both  conditions. 

The  members  in  Fig.  70  are  identical  with  those  of  Fig.  69, 
and  their  connections  with  each  other  are  the  same  as  before.  If 
the  member  #,  the  original  connecting-rod,  be  made  the  fixed  mem- 
ber, as  in  Fig.  71,  a  and  d  are  both  moving  members;  but  they 
still  rotate,  relatively ,  about  Oad.  This  mechanism,  as  shown  in 
Fig.  71,  is  that  of  the  oscillating  steam-engine,  in  which  a  corre- 
sponds to  the  crank,  b  to  the  frame,  c  to  the  cylinder,  and  d  to  the 
piston-rod  and  piston.  Fig.  72  represents  another  condition  of 
this  same  mechanism,  in  which  c,  the  original  crosshead,  is  the 
fixed  member.  Under  any  of  these  four  conditions  the  relative 

*  The  notation  used  in  the  following  discussion  is  this  :  small  letters,  a,  b, 
e,  etc.,  are  used  to  designate  the  different  members  ;  0  is  used  for  all  centres 
(instant  or  permanent),  and  the  subscripts  of  0  indicate  the  members  which 
rotate  relatively  about  it.  Thus  Oac  indicates  that  the  member  a  rotates  rela- 
tive to  c  (or  c  relative  to  a)  about  the  point  designated  as  Oa«> 


TRANSMITTING  MOTION  IN  MACHINES. 


61 


motion  of  a  to  d  (or  of  d  to  a)  is  a  simple  rotation  about  the  centre 
Oad.  In  a  similar  way,  the  relative  motion  of  a  and  ~b  is  a  rotation 
about  the  centre  Oab\  that  of  Z>  and  c  is  a  rotation  (or  oscillation) 


Fig.  71 


Fig.  72 


about  Obc ;  that  of  c  and  ^  is  a  translation  parallel  to  the  centre 
line  of  d,  or  a  rotation  about  a  centre,  Oc<f,  lying  at  infinity  and  in 
such  a  line  as  NN. 

It  is  evident  from  the  above  illustration  that  apparently  very 
different  mechanisms  may  be  essentially  the  same  thing  under  dif- 
ferent conditions.  This  was  referred  to  in  speaking  of  the  classifi- 
cation of  the  parts  of  machines  in  Art.  24. 

Such  changes  in  the  condition  of  a  mechanism  as  are  illustrated 
in  Figs.  69,  70,  71,  and  72,  and  which  are  effected  by  making  dif- 
ferent members  correspond  to  the  stationary  part,  or  frame,  are 
called  by  Professor  Reuleaux  The  Inversion  of  Mechanisms. 

Other  examples  of  inversion  will  be  given  in  a  later  part  of  this 
work. 

40.  Relative  Motion  between  Different  Members  of  a  Mechan- 
ism.— The  relative  motions  of  the  adjacent  members  of  a  mechan- 
ism are  usually  comparatively  simple;  thus  in  the  mechanism  of 
Fig.  73  the  relative  motions  of  a  to  #,  b  to  c,  c  to  d,  and  a  to  d  are 
simply  rotations  or  oscillations  about  permanent  centres.  The  rela- 
tive motions,  in  Fig.  69,  of  the  corresponding  adjacent  members 
were  traced  in  the  preceding  article. 

In  any  mechanism  each  link  (member)  has  a  distinct  motion 
relative  to  each  of  the  other  members.  In  four-piece  mechanisms, 


62  KINEMATICS  OF  MACHINERY. 

as  those  of  Figs.  69  and  73,  there  are  six  distinct  relative  motions, 
viz.  :  a  to  5,  1)  to  c,  c  to  d,  a  to  d,  a  to  c,  and  Z>  to  d.  Each  of  these 
motions  is  equivalent  to  rotation  or  oscillation  about  a  centre  (per- 
manent or  instant).  Four  of  these  are  permanent  centres  in  the 
mechanisms  referred  to  above,  viz.  :  Oab,  06c,  Ocd,  and#ad.* 

In  Fig.  73,  OK  rotates  relative  to  d  in  an  arc  of  finite  radius. 
In  Fig.  69,  the  motion  of  Obc  relative  to  d  is  equivalent  to  rotation 
about  the  centre  0^,  in  an  arc  of  infinite  radius.  If  it  were  pos- 
sible to  supply  a  link  of  infinite  radius  connecting  the  point  Ocd  of 


d  with  the  point  Obc,  the  mechanism  of  Fig.  69  would  be  similar 
in  character  to  that  of  Fig.  73;  or  the  former  may  be  considered 
as  a  limiting  form  of  the  latter.  The  practical  mechanism  of 
Fig.  69  is  the  exact  kinematic  equivalent  of  such  an  imaginary 
mechanism. 

The  relative  motions  of  the  opposite  links,  a  and  c,  and  b  and  d, 
of  Figs.  69  and  73,  are  not  so  evident  as  are  the  motions  of  the 
adjacent  members;  but  the  instant  centres  for  these  motions  are 
readily  located  from  the  principles  of  Art.  19.  In  the  first  place 
it  is  to  be  noted  that  the  instant  centre  for  two  members  is  a 
common  or  coincident  point  of  both,  for  it  is  a  point  with  regard 
to  which  neither  of  them  has  any  motion.  If  this  point  lies 

*  In  the  mechanism  of  Fig.  69  the  centre  Ocd  is  at  infinity.  While  it  is  not 
an  actual  physical  pin  like  the  other  permanent  centres,  still  it  is  properly  a 
permanent  centre  rather  than  an  instant  centre,  because  it  is  equivalent  to  a 
fixed  centre  of  a  link  of  infinite  length. 


TRANSMITTING  MOTION  IN  MACHINES.  63 

outside  of  either  actual  bod}*,  this  body  may  be  imagined  as 
extended  to  include  this  centre,  for  a  body  may  have  rigid  con- 
nection with  any  point  relative  to  which  it  has  no  motion,  as  stated 
in  Art.  5. 

It  is  to  be  borne  in  mind,  then,  that  Oab  is  a  point  in  both  a 
and  b\  Oac  is  a  point  in  both  a  and  c,  etc.  This  enables  us  to  locate 
the  instant  centres  for  the  opposite  links.  In  Fig.  73  Oo6,  as  a 
point  in  a  rotating  relative  to  d,  must  move  in  a  line  perpendicular 
to  the  line  joining  the  points  Oad-0ab;  hence  its  motion  is  equiva- 
lent to  a  rotation  about  some  point  in  this  line  or  its  extension 
(see  Art.  19).  Likewise,  the  motion  of  Obc  relative  to  d  is  equiv- 
alent to  a  rotation  about  some  point  in  the  line  Ocd-0bc  or  its 
extension.  But  Oab  and  Obc  are  two  points  in  the  link  6,  and,  as 
such,  must  have  the  same  motion,  i.e.,  rotation  about  the  point 
Obd  at  the  intersection  of  the  lines  Oad-0ab  and  Ocd-0bd.  The 
motion  of  these  two  points  determines  the  motion  of  the  link. 
Therefore  Obd  is  the  instant  centre  for  the  motion  of  b  relative 
to  d.  In  a  similar  way,  all  points  of  b  rotate  relative  to  a  about 
the  centre  Oab,  and  all  points  of  d  rotate  relative  to  a  about  Oad. 
The  points  Obc  and  Ocd  are  points  in  b  and  d,  respectively;  hence 
they  move,  relatively  to  a,  perpendicularly  to  the  lines  Obc-0ab 
and  Ocd-0ad  respectively,  and  the  intersection  of  these  lines,  or 
Oacj  is  their  common  centre  of  rotation  relative  to  a.  But  Obc 
and  Ocd  are  two  points  in  the  link  c;  therefore  the  point  Oac  is 
the  instant  centre  for  the  motion  of  c  relative  to  a.  We  have 
thus  located  all  of  the  instant  centres  for  the  mechanism  of  Fig.  73. 

Four  of  the  centres  for  the  mechanism  of  Fig.  69  have  been 
located,  viz.:  the  permanent  centres  Oad,  Oab,  Obc,  and  Ocd  (the 
last  at  infinity).  The  centres  for  the  two  pairs  of  opposite  links, 
b  and  d,  and  a  and  c,  are  yet  to  be  found.  The  former,  Obd  is 
readily  found;  for  Oab  (a  point  in  6)  moves  perpendicularly  to 
the  centre  line  of  a,  and  Obc  (also  a  point  in  b)  moves  perpendicu- 
larly to  NN;  therefore  the  intersection  of  these  lines,  or  0M,  is 
the  required  instant  centre  for  the  motion  of  6  relative  to  d. 

The  reasoning  by  which  the  centre  for  the  relative  mofion  of  a 
and  c  is  found  is  somewhat  more  involved.  The  point  Obc  is  a  point 
common  to  b  and  c.  AH  points  in  6  rotate  relative  to  a  about  the 


64  KINEMATICS  OF  MACHINERY. 

centre  0ab;  therefore  Obc  as  a  point  in  b  rotates  about  this  point, 
or  it  moves,  relative  to  a,  perpendicularly  to  the  centre  line  of  b 
(the  line  Obc-0ab).  The  point  Ocd  (at  infinity)  is  a  point  common 
to  c  and  d.  As  a  point  in  d  it  rotates  about  Oad  relative  to  a, 
moving  perpendicularly  to  a  vertical  line  through  Oad,  or  to  N'  N'. 
One  point  of  c  (Obc)  moves  perpendicularly  to  Obc-0ab,  and  another 
point  of  c  (Oca)  moves  perpendicularly  to  N' N' \  hence  the  instant 
centre  for  the  motion  of  c  relative  to  a  is  at  the  intersection  of 
these  two  lines,  or  at  the  point  0^. 

By  considering  c,  instead  of  a,  as  the  stationary  member,  the 
same  result  may  be  reached  rather  more  easily. 

It  is  possible  to  locate  all  of  the  instant  centres  of  a  mechanism 
of  four  members  by  the  principles  already  given;  but  with  higher 
numbers  of  members  this  cannot  always  be  done,  and  a  very  im- 
portant theorem  given  by  Professor  Kennedy  affords  a  ready  solu- 
tion in  these  more  difficult  cases.  This  theorem  is  often  advan- 
tageous even  in  four-link  mechanisms,  and  by  its  aid  the  rather 
tedious  reasoning  employed  above  in  finding  Oac  for  the  mechanism 
of  Fig.  69  can  be  avoided. 

The  statement  of  this  theorem,  as  given  by  Professor  Kennedy, 
is :  "  If  any  three  bodies  a,  1),  and  c  have  plane  motion,  their  vir- 
tual \instant\  centres  Oab,  Obc,  and  Oac  are  three  points  upon  one 
straight  line." 

This  theorem  applies  to  any  three  bodies  having  plane  motion, 
whether  they  be  members  of  the  same  mechanism  or  not.  Two  of 
the  three  centres  being  known,  or  assumed,  the  following  demon- 
stration proves  that  no  point  lying  outside  of  the  line  connecting 
the  known  centres  can  be  the  required  third  centre;  hence  this 
third  centre  must  lie  in  the  line  connecting  the  other  two,  as  stated 
in  the  theroem. 

In  Fig.  74,  let  a,  b,  and  c  be  any  three  bodies  moving  in  a 
plane,  members  of  a  single  mechanism  as  indicated  by  the  heavy 
lines,  or  entirely  independent  bodies. 

Suppose  that  a  rotates  relative  to  b  about  the  centre  Oab,  and  that 
c  rotates  relative  to  b,  about  the  centre  Obc.  Then  0aB  is  a  point 
common  to  a  and  b,  and  Obc  is  a  point  common  to  b  and  c.  As- 
sume that  such  a  point  as  0'  is  the  instant  centre  for  the  relative 


TRANSMITTING   MOTION  IN  MACHINES.  65 

motion  of  a  and  c;  then  this  point  is  a  common  point  of  a  and  c. 

All  points  in  a  must  rotate  relative  to  b  about  Oa&,  and  all  points 

in  c  must  rotate  relative  to  b 

about  Obc-     As  a  point  in  a  the 

motion  of  0'  relative  to  b  must  be 

in  a  direction  perpendicular  to 

Oab-0f,  while  as  a  point  in  c  its 

motion  relative  to  b  must  be  in  a 

direction  perpendicular  to  0&c— 0'. 

A  point  can  have  but  one  motion 

relative  to  a  given  body  at  any 

time,  and  therefore  these  perpendiculars  must  coincide.     This  is 

possible  only  when  Oab-0'  and  0&c-0'  are  in  one  straight  line, 

viz.:    the  line  joining  the  given  centre  Oa&  and  0&c.     It  is  thus 

seen  that  the  point  0'  cannot  be  the  centre  required,  unless  it 

does  lie  in  such  line. 

This  theorem  does  not  locate  the  centre  Oac  definitely;  for  it 
may  be  any  place  along  the  line  of  Oab-0bc,  between  these  centres, 
or  beyond  either  of  them.  This  is  as  it  should  be,  for  in  the 
arrangement  of  Fig.  74  there  is  no  prescribed  connection  between 
a  and  c,  and  their  relative  motion  is  therefore  not  definitely  con- 
strained.* 

If  a  fourth  member,  d,  which  will  constrain  the  motion  of  a 
relative  to  c — such  as  a  link  connecting  the  free  ends  of  a  and  c 
— be  introduced,  another  combination  of  three  members  (as  a,  c, 
and  d)  may  be  taken,  in  which  Oad  and  Ocd  are  known,  and,  by 
the  theorem,  the  other  centre  for  this  combination  (Oac)  will  lie 
in  the  line  of  the  known  centres.  In  this  constrained  four-link 


*  It  is  to  be  noticed  that  the  theorem  discussed  above  has  reference  to 
three  members,  and  that  these  three  members  involve  three  instant  centres  ; 
any  member  has  a  centre  with  reference  to  each  of  the  other  members.  In 
a  combination  of  three  bodies  every  letter  which  stands  for  one  body  is  used 
twice  as  a  subscript  to  0.  If  two  of  the  three  centres  are  given  their  symbols 
will  have  one  common  letter  in  their  subscripts,  and  the  third  (required)  centre 
•will  have  for  a  subscript  the  two  odd  letters.  Thus  if  Oab  and  Obc  are  the 
given  centres,  Oac  is  the  third.  This  is  a  convenient  aid  in  applying  the  above 
theorem  to  a  mechanism. 


66 


KINEMATICS  OF  MACHINERY. 


mechanism  there  are  two  lines,  each  of  which  contains  Oac  (viz.: 
Oab-0bc,  and  Oad-0cd)',  hence  Oac  is  at  their  intersection,  or  its 
position  is  definitely  determined  (see  Fig.  73).  . 

By  referring  to  Figs.  69  and  73  it  will  be  seen  that  the  loca- 
tions of  the  centres,  as  already  determined,  agree  with  the  state- 
ment of  the  theorem.  Thus,  as  to  the  links  a,  b,  and  c,  the 


Oef  at  ±  oo 


at  intersection 
Fig.  75 


centres  Oa&,  0&c,  and  Oac  lie  in  one  line;  also,  as  to  a,  c,  and  d, 
Oac,  Oad,  and  Ocd  lie  in  one  line,  and  Oac  lies  at  the  intersection 
of  these  two  lines. 

As  an  illustration  of  the  application  of  the  above  theorem  to 
more  than  four  members,  the  mechanism  of  a  common  type  of 
crank  shaper,  indicated  in  Fig.  75  by  what  is  called  a  skeleton 
drawing,  may  be  taken. 


TRANSMITTING  MOTION  IN  MACHINES. 


67 


In  this  machine  there  are  six  members:  the  cranK,  a;  the 
sliding  block,  6;  the  vibrator,  c;  the  link,  d;  the  ram,  e\  and  the 
frame,  /,  to  which  the  members  a  and  d  are  pivoted,  and  which 
is  provided  with  guides  for  the  motion  of  e.  This  mechanism  has 
15  instant  centres,*  of  which  7  are  also  permanent  centres.  Of 
the  permanent  centres,  Oaf,  Oa&,  Ocd,  Oce,  and  Odf  are  located 
at  the  points  of  connection  of  adjacent  members,  while  0&c  and 
Oef  are  at  infinity.  The  other  centres  are  found  by  the  use  of 
.Kennedy's  theorem. f 

The  following  scheme  suggests  the  solution  of  this  problem : 


Centre  Required. 

Lies-at  Intersection  of  the  Lines. 

Ocf 
Oae 
Oac 

Obf 

Oae 
Oad 
Obd 
Obe 

Ocd  —  Odf  and  Oce  —  Oef 

Ocd  —  Oce       "       Odf—  Oef 
Oab—Obc       "       Ocf  —  Oaf 
Oab—Oaf       "       Obe—  Ocf 
Oac—  Oce       "      Oaf—  Oef 
Oaf—  Odf       "       Oac  —  Ocd 
Oab—Oad      "       Ocd  —  Obe 
Obf—  Oef       "       Obe—  Oce 

The  method  of  instant  centres  will  be  frequently  used  in  the 
iater  part  of  this  work,  especially  in  treating  linkwork  ;  but  it  may 
be  well  to  give  an  illustration  of  its  use  at  the  present  place.  It  is 
to  be  remembered  that  the  linear  velocity  of  a  point  which  is  mov- 
ing relative  to  any  body  is  proportional  to  its  distance  from  the 
centre  about  which  it  rotates  relative  to  that  body.  In  Fig.  69, 
for  example,  the  body  a  rotates  relative  to  the  stationary  member, 


*  In  a  mechanism  of  n  members,  there  are instant  centres, 

2 


some 


of  which  are  also  permanent  centres. 

f  A  diagram  such  as  is  shown  in  connection  with  Fig.  75  is  convenient  in 
this  work.  The  unshaded  spaces  indicate  the  centres  to  be  located,  and  the 
memory  is  aided  by  checking  off,  in  the  proper  place,  each  centre  as  it  is  found. 


68  KINEMATICS  OF  MACHINERY. 

d  (the  frame),  about  Oad.  If  the  linear  velocity,  vt,  of  Oab  is 
known,  the  linear  velocity,  t>9,  of  the  crosshead  (piston)  is  readily 
found  by  the  principles  of  instant  centres.  The  point  06c  is  com- 
mon to  the  connecting-rod,  #,  and  the  crosshead,  c\  while  the  point. 
Oab  is  common  to  the  crank,  a,  and  the  connecting-rod,  Z».  The 
instant  centre  of  b  relative  to  d  is  0M;  then,  as  all  points  of  b  must 
have  the  same  angular  velocity  about  0M  their  linear  velocities  are 
proportional  to  their  distances  from  this  centre  ;  hence 

v9  :  vl  : :  Ob(I-Obc  :  Obd-0ab. 

If  v1  be  laid  off  from  Oab  toward  Obd,  along  the  line  connecting 
these  points,  and  then  a  line,  mn,  parallel  to  the  connecting-rod, 
be  drawn  till  it  cuts  the  normal  NN,  the  length  on  this  normal 
from  Obc  to  n  equals  va,  from  the  above  proportion.  As  the  motion  of 
c  relative  to  d  is  a  translation,  all  points  of  c  have  the  same  velocity 
relative  to  d ;  hence  v^  is  the  velocity  of  the  crosshead,  or  piston,, 
relative  to  the  frame,  or  cylinder. 

In  an  engine  the  crank  rotates  about  the  shaft  with  a  velocity 
which  is  usually  taken  as  uniform  ;  while  the  velocity  of  the  cross- 
head  (or  piston)  is  variable.  The  velocity  of  the  piston  can  be 
found  for  any  phase  by  laying  off  the  crank-pin  velocity  along  the 
extension  of  the  crank,  drawing  a  line  (as  mn,  Fig.  69)  parallel  to- 
the  connecting-rod  till  it  cuts  the  normal  (NN)  through  the  cross* 
head  pin. 

A  modification  of  the  preceding  construction  is  often  eveiL 
more  convenient.  Lay  off  the  line  N'-N'  (Fig.  69)  through 
the  centre  of  the  shaft,  Oad,  perpendicular  to  the  line  of  the 
piston  travel.  The  connecting-rod  (extended  if  necessary)  cuts 
N'-N'  in  the  point  Oac,  then,  since  v2 :vl:: Obd-0bc : Obd-0ab  and, 
from  similar  triangles,  Obd-0bc:0bd-0ab::0ad-0ac:0ad-0ab,  it  fol- 
lows that  v2:v1:  :0ad-0ac'  'Oad-0ab. 

It.  appears  from  this  proportion  that  when  the  length  of  the 
crank,  Oad-0ab,  is  taken  to  represent  the  uniform  crank-pin 
velocity,  the  cross-head  velocity  is  represented  by  the  distance, 
Oa<j-0ac,  the  intercept  on  the  perpendicular,  N'-Nf,  between 


TRANSMITTING  MOTION  IN  MACHINES. 


69 


the  shaft  centre  and  the  line  of  the  connecting-rod,  the  latter 
extended  if  necessary. 

This  relation  is  also  evident  from  the  consideration  that  the 
instant  centre,  Oac,  as  a  common  point  of  a  and  c,  has  the  same 
velocity  and  direction  of  motion  in  c  as  in  a.  As  a  point  of  c 
the  linear  velocity  of  Oac  is  v2,  since  all  points  of  c  have  the  same 
velocity  of  translation.  This  velocity  is  found  from  the  motion 
of  Oac  as  a  point  in  a  by  the  proportion  v2:vl:  :0ad-0ac:0ad-0ab. 

In  general,  when  the  instant  centres,  Oa&,  Oac,  and  0&c,  for 
the  plane  motion  of  any  two  bodies,  a  and  6,  relative  to  each 
other  and  to  a  third  (reference)  member,  c,  are  located,  the  linear 
velocity  of  any  point  in  6  corresponding  to  a  given  linear  velocity 


Fig.  75a 

of  any  point  in  a  can  be  found  graphically.  In  Fig.  75a,  let  vt 
represent  the  given  linear  velocity  of  any  point,  P,  in  a,  and  let 
the  corresponding  velocity,  v2,  of  any  point,  Q,  in  6,  be  required. 
The  linear  velocity,  v1 ',  of  Oab  as  a  point  of  a  is  found  from  the 
proportion— v' :  vl : :  Oac-0ab  •  Oac-P,  by  the  construction  shown. 
Using  Oab  as  a  point  of  6,  the  corresponding  linear  velocity  of  Q 
is  found,  by  a  similar  construction,  from  the  proportion, 
v2:vf::Obc-Q:Obc-Oab. 

The  ratio  of  the  angular  velocities  of  any  two  bodies,  a  and  b, 
having  plane  motion  relative  to  a  third  body,  c,  may  also  be 
determined  when  the  three  instant  centres  are  located.  Let  ojv 
and  w2  be  the  respective  angular  velocities  of  a  and  b  relative  to 
c  in  Fig.  75a.  Since  Oab  as  a  common  point  of- a  and  6,  has  a 


70 


KINEMATICS  OF  MACHINERY. 


linear  velocity  i/,  the  angular  velocities  of  a  and  6  relative  to  c 
are  equal  to  this  linear  velocity  divided  by  the  respective  instant 
radii.  That  is,  wl  =v' -r-0ab-0ac  and  aj2=v'  +  0ab-0bc.  Hence 
(1)^:0)2-  -Oab-Obc'-Oab-Oac-  When  this  proportion  is  used  in  the 
case  of  any  mechanism  the  resulting  value  of  the  angular  velocity 
ratio  is  identical  with  that  obtained  by  the  methods  of  Arts.  29-32. 
41.  Velocity  Diagrams. — It  has  been  shown  in  the  preceding 
article  how  the  method  of  instant  centres  can  be  used  to  determine 
the  linear  velocity  of  one  point  from  the  known  velocity  of  another 
point.  It  is  often  desirable  to  represent,  graphically,  the  velocities 
of  a  point  at  various  phases  of  a  mechanism,  and  this  is  done  con- 
veniently by  velocity  diagrams.  Fig.  76  shows  the  mechanism  of 


Fig.  76  (a) 

the  reciprocating  engine  in  outline.  C  is  the  crank-pin,  c  is  the 
crosshead-pin,  Q  is  the  centre  of  the  shaft.  The  crosshead  moves 
from  0  to  9  and  back  again  to  0  during  one  complete  rotation  of 
the  crank.  The  simultaneous  positions  of  crosshead-pin  and  crank- 
pin  are  indicated  respectively  by  0,  1,  2,  3,  etc.,  and  0',  V,  2',  3', 
etc.  As  shown  in  the  preceding  article,  if  the  linear  velocity  of  the 
crank-pin  is  represented  by  the  length  of  the  crank,  r,  the  velocity 
of  the  crosshead  for  any  phase  is  represented  by  the  segment,  s,  of 


s 


TRANSMITTING  MOTION  IN  MACHINES. 


71 


the  line  N-N,  which  lies  between  Q  and  the  line  of  the  connecting 
rod,  C-c.  If  the  segment,  s,  is  found  for  each  of  the  crosshead 
positions  from  0  to  9,  the  corresponding  lengths  of  s  may  be  erected 
as  ordinates  to  0-9  at  the  corresponding  crosshead  positions.  A 
curve  passing  through  the  upper  ends  of  these  ordinates  gives  a 
velocity  diagram  of  the  point  c  with  the  path,  0-9,  as  a  base. 
This  diagram  is  called  a  Velocity-Space  Diagram.  If  a  sufficient 
number  of  ordinates  have  been  determined  'the  diagram  gives 
quite  accurately  the  velocity  of  c  for  intermediate  positions. 
Fig.  77  shows  a  method  of  constructing  a  velocity  diagram  upon 


Fig.  77 


a  curved  path  as  a  base.  The  driving  arm,  or  crank,  a,  imparts, 
by  its  rotation,  a  reciprocating  motion  to  the  arm  c  in  the  arc  1-9. 
The  point  Obc  occupies  the  positions  1,  2,  3,  etc.,  when  the 
point  Oab,  is  at  the  corresponding  points  1' ',  2',  3',  etc.  If  2'-2/ 
is  laid  off  equal  to  the  linear  velocity  of  Oa&  upon  the  extension  of 


72  KINEMATICS  OF  MACHINERY. 

the  line  of  a,  and  2/-2J  is  drawn  parallel  to  6,  the  segment  of  the 
extension  of  c  cut  off  by  this  parallel  equals  the  linear  velocity  of 
the  point  Obc.  This  is  proven  by  reference  to  the  instant  centre 
of  b  and  d,  Obd;  for  the  linear  velocity  of  Oab  relative  to  d  is  to 
the  velocity  of  Obc  as  Obd-0ab  is  to  Obd-0bc  (the  linear  velocities  of 
two  points  in  6  relative  to  d  are  proportional  to  their  radii  from 
Obd).  But  2'-2/  (the  velocity  of  Oab)  is  to  2-2t  as  Obd-0ab  is  to 
Obd-0bc,  and  therefore  2-2t  is  the  velocity  of  Obc.  By  a  similar 
construction  for  other  phases,  the  corresponding  velocities  of  the 
point  Obc  may  be  obtained.  If  these  velocities  of  the  driven  point 
are  laid  off  as  radial  ordinates  at  the  corresponding  points  in  its 
path,  the  curve  1^-2^-3^  etc.,  may  be  drawn,  and  it  is  the  velocity 
diagram  of  Obc  on  its  path  as  a  base.  This  is  called  a  Radial 
Velocity  Diagram. 

If  the  motion  of  the  driving  point,  Oab,  is  uniform,  its  velocity 
diagram  is  a  circle  concentric  with  its  path,  as  drawn  in  Fig.  77, 
but  the  method  applies  equally  well  if  the  driving  point  has  a 
variable  velocity.  A  velocity  diagram  with  rectangular  co-ordi- 
nates may  be  constructed  from  the  one  just  determined  by  rectify- 
ing the  path  of  Obc,  1-2,  etc.,  and  erecting,  at  the  various  points, 
parallel  ordinates  of  lengths  found  as  above.  This  derived  velocity 
diagram  is  shown  in  Fig.  77a,  but  it  is  seldom  necessary  to  con- 
struct it. 

If  on  various  positions  of  the  crank  (Fig.  76)  the  corresponding 
velocities  of  the  follower,  c,  are  laid  off  radially  from  Q,  as  Q-l", 
Q-2",  etc.,  and  a  curve  is  then  drawn  through  I",  2",  3",  etc.,  a 
Polar  Velocity  Diagram  of  the  motion  of  c  is  obtained.  This  is 
sometimes  preferred  to  the  rectangular  diagram  on  the  path  of  the 
follower. 

It  is  often  desirable  to  show  the  relation  between  velocity  and 
time.  For  this  purpose  a  diagram  may  be  constructed  (Fig.  76a) 
in  which  ordinates  represent  velocity  and  abscissas  represent  time. 
This  is  called  a  Velocity  Time  Diagram. 

In  the  illustrations  of  Arts.  39,  40,  and  41,  linkwork  mech- 
anisms have  been  taken,  as  the  methods  developed  in  these  arti- 
cles are  especially  useful  in  the  treatment  of  this  class;  but  the 
deductions  are  also  applicable  to  other  mechanisms. 


TRANSMITTING  MOTION  IN  MACHINES. 


73 


42.  Acceleration  Diagrams.— In  any  velocity-space  diagram 
the  subnormal  to  the  curve  at  any  point  is  proportional  to  the 
corresponding  acceleration.  When  different  scales  are  used  for 
velocity  (ordinates)  and  for  space  (abscissas),  as  is  usually  the 
case,  still  another  scale  must  be  used  for  acceleration. 

Let  OPQ,  Fig.  78,  be  any  velocity-space  curve  in  which  I" 
of  ordinate  represents  n  times  as  many  velocity  units  (ft.  per 
sec.)  as  1"  of  abscissa  represents  space  units  (ft.). 

Let  PM,  PT,  and  PN  be  the  respective  ordinate,  tangent  and 


Fig.  78a 


normal  to  the  curve  at  any  point,  P,  and  let  6  be  the  angle  between 
the  tangent,  PT,  and  the  base  line,  TN. 

ds 
v  =—  =nPM  =  velocity  represented  by  the  ordinate  PM,  when 


the  length  PM  is  measured  by  the  space  scale; 


dv 


p=-r  =  corresponding  acceleration; 
at 

PM     MN     dv 
=        =  ~PM 


7ivdt' 


Hence  the  subnormal,  MN,  is  proportional  to  the  acceleration 
and  may  be  used  as  an  ordinate,  at  M,  of  an  Acceleration  Space 
Diagram  (see  also  Fig.  76).  When  so  used  1"  of  ordinate  repre- 
sents n2  times  as  many  acceleration  units  (ft.  per  sec.2)  as  1" 
of  abscissa  represents  space  units  (ft.).  Since  1"  of  velocity 


74  KINEMATICS  OF  MACHINERY. 

ordinate  represents  n  times  as  many  velocity  units  as   1"  of 
abscissa  represents  sp^ace  units,  1"  of  accleration  ordinate  repre- 

n2 
seats  —  =n  times  as  many  acceleration  units  as  V  of  velocity 

n 

ordinate  represents  velocity  units. 

The  engine  mechanism  of  Fig.  76  is  drawn  to  a  scale  of  J  size.* 
The  ordinates  of  the  velocity  curve  represent  the  velocity  of  the 
cross-head  to  a  scale  on  which  the  length  of  the  crank  on  the 
drawing  measures  the  linear  velocity  of  the  crank-pin.  Taking 
this  velocity  as  9  ft.  per  sec.  and  the  actual  length  of  the  crank 
as  9"  (represented  on  the  drawing  by  9"Xi  =  lJ");  the  velocity 
scale  is  1 J"  =9  ft.  per  sec.  or  V  =8  ft.  per  sec.  The  space  scale 
is  1"=8",  or  1"=J  ft.  .-.  w=8-f-§=12.  The  acceleration  scale 
is  l"=n2Xf  =144X  J=96  ft.  per  sec.2 

In  any  velocity-time  diagram  the  slope  of  the  tangent  to  the 
curve  at  any  point  is  proportional  to  the  acceleration.  Fig.  78a 
shows  a  method  of  constructing  an  Acceleration-Time  Diagram. 
PM  and  QH  are  two  ordinates  of  any  velocity-time  curve  OPQ, 
at  any  convenient  distance  apart,  TK  is  tangent  to  OPQ  at  P, 
and  makes  an  angle,  6,  with  the  base  line,  TH .  QH  is  extended 
to  cut  TK  at  K,  and  PG  is  drawn  parallel  to  TH.  From  M  take 
ML=KG  as  an  ordinate  of  the  acceleration  curve,  and  determine 
other  ordinates  in  the  same  way,  the  distance  between  the  two 
ordinates  used  being  equal  to  PG  in  each  case. 

In  Fig.  78a,  p  =  — -  =  tan  0  =  — -.     Therefore  KG  measured  in 
at  r(j 

velocity  units  is  the  acceleration  in  corresponding  acceleration 
units  during  an  amount  of  time  represented  by  the  length  of  PG. 

When  PG  =  —  sec.,  p  =mKG.     On  the  corresponding  acceleration 
m 

scale  I"  represents  m  times  as  many  acceleration  units  (ft.  per  sec.2) 

as  1"  on  the  velocity  scale  represents  velocity  units  (ft.  per  sec.). 

Using  the  same  data  as  in  the  preceding  example,  i.e. :  velocity 

crank-pin  =  9   ft.   per   sec.,   and   length   of   crank  =  9"  =  f   foot, 

the  crank  rotates =  1.91  times  per  sec.     The  time  of  1  rev. 

t  X2;r 

*  The  figure  is  reduced  to  about  £  size  in  reproduction. 


TRANSMITTING  MOTION  IN  MACHINES.  75 


is  -  =.52  sec.     This  time  is  divided  into  eighteen  equal  parts 
i  .y  i 

in  constructing  the  diagram  in  Figs.  76  and  76a,  and  two  of  these 
parts  are  used  as  the  distance  between  ordinates  in  constructing 
the  acceleration-time  diagram.  These  two  parts  represent 

52x2  1 

sec.  =  .058  sec.     m  =  —  -  =17.3.     The  acceleration  scale  is 


18 
I"  =8X  17.3  =138  fi.  per  sec.2 

\a  the  preceding  discussion  the  foot-second  system  of  units 
was  used  throughout.  Any  other  system  of  units  may  be  used 
in  a  similar  manner  and  the  corresponding  scales  determined  by 
the  same  methods. 

It  may  be  noted  that  the  acceleration  is  indeterminate,  graphic- 
ally, on  the  velocity-space  diagram,  where  the  curve  crosses  the 
axis  of  Jf.  It  can  be  found  for  several  ordinates  near  that  point  and 
extended  to  the  end  position  without  much  error.  On  the  time- 
velocity  diagram,  however,  it  is  wholly  determinate.  Both  methods 
are  open  to  the  objection  that  considerable  error  is  necessarily  in- 
troduced in  drawing  tangents  to  curves  which  are  not  very  well 
defined  themselves. 

If  the  weight  of  the  moving  body  is  known  the  force  required 
to  accelerate  or  retard  it  at  any  position  can  be  found  from  the 
acceleration  curve.  If  F~be  this  force,  W  the  weight  of  the  body,  and 

p  the  acceleration,  F  =  -  —  p.     The  acceleration  can  be  read  off 

from  the  acceleration  scale  at  any  point  and  the  force  corresponding 

W 

may  be  found  simply  by  multiplying  the  acceleration  by  —  .     Or 

y 

a  force  scale  may  be  constructed,  as  can  readily  be  seen. 

43.  Centrodes  and  Axodes,  —  The  instant  centre  for  two  bodies 
having  plane  motion  may  also  be  a  permanent  centre,  in  which 
case  it  remains  a  fixed  point  in  both  bodies  ;  but  in  the  general  case 
the  instant  centre  does  not  occupy  the  same  position  in  either  body 
for  any  two  successive  relative  positions  of  these  bodies,  and  the 
locus  of  the  instant  centre  upon  each  of  the  bodies  is  called  a  Cen- 
trode.  The  instant  centre  is  a  point  common  to  the  two  bodies  for 
the  instant,  and  therefore  the  two  coincident  points  of  the  bodies 
which  lie  at  this  centre  have  for  the  instant  no  relative  motion  ;  but 


76 


KINEMATICS  OF  MACHINERY. 


any  other  two  points  (one  in  each  of  these  bodies)  do  move  rela- 
tively. The  pair  of  centrodes  traced  on  the  bodies  by  the  motions 
of  the  instant  centre  are  tangent  to  each  other  at  the  instant  centre; 
and  as  these  contact  points  of  the  centrodes  have  no  relative  motion, 
the  pair  of  centrodes  roll  on  each  other  with  a  pure  rolling  action. 
Points  in  the  pair  of  centrodes  which  previously  coincided  in  the 
instant  centre  are  now — in  common  with  other  points  of  the  two 
bodies — rotating  relatively  about  the  present  instant  centre;  and  a 
similar  remark  applies  to  a  pair  of  such  points  which  may  coincide 
in  the  instant  centre  at  any  succeeding  phase. 

Any  plane  motion  between  two  bodies,  whatever  the  mechan- 
ism adopted  for  producing  this  motion,  is  exactly  equivalent  to 
that  resulting  from  the  rolling  upon  each  other  of  two  members 
whose  contact  lines  conform  to  the  centrodes  for  this  motion. 
The  nature  of  this  action  may  be  made  clearer  by  reference  to 
Fig.  79.  Let  a  and  b  be  two  points  in  the  body  -4,  moving  rela- 


Fig.  79 

tive  to  the  fixed  body  M  so  as  to  occupy  in  succession  the  positions 
a-b,  a'-b',  a"-b",  etc.  From  the  middle  of  a-a'  and  b-b'  erect  per- 
pendiculars to  these  lines,  intersecting  in  0  ;  then  the  motion  from 
a-b  to  a'-b'  is  equivalent  to  a  rotation  about  the  point  0  as  a  centre 
through  the  angle  aOa'  =  bOb'  =  (p.  The  motion  a'-b'  to  a"-b" 
is  likewise  equivalent  to  a  rotation  about  0'  through  the  angle  0', 
etc.  0,  0',  etc.,  are  temporary  or  shifting  centres,  and  they  may 
be  connected  by  the  polygon  0-0'- 0",  etc.,  which  lies  in  the  station- 
ary body  M.  If  the  line  O-O/  be  laid  off  on  the  moving  body  A  of 


TRANSMITTING  MOTION  IN  MACHINES.  77 

a  length  equal  to  0-0'  and  making  the  angle  0  with  the  latter,  it 
is  evident  that  when  a-b  moves  to  a'-b' ',  0-0 '/  will  fall  along  0-0' 
and  0'  will  coincide  with  0'. 

From  0'  lay  off  an  angle  with  O'-O"  equal  to  0';  extend  0-0' 
to  #  through  this  last  angle;  and  let  the  angle  gO'O"  =  /?'. 
This  extension  divides  0'  into  /?'  and  a'v  and  a'  =  0'  —  </?'. 
Extend  O-O/  to  the  right;  from  O/  lay  off  the  line  0/-0,"  equal 
to  O'-O"  and  making  an  angle  with  O-O/  equal  to  a'.  When  a'-b' 
has  moved  to  «"-£",  O/'  will  coincide  with  0".  By  a  continuation  of 
this  process  the  polygon  O-Oi'-O/'-O/",  etc.,  is  constructed  on  the 
face  of  the  moving  body  A ;  and  the  given  motion  of  A  (a-b,  a'-b', etc.) 
is  equivalent  to  the  broken  rolling  action  of  this  polygon  of  A  upon 
the  polygon  previously  formed  on  M.  If  the  positions  of  A  (a-b, 
a'-b',  etc.)  are  taken  closer  together,  the  corresponding  positions  of 
the  temporary  centres  (0,  0',  etc.)  become  closer,  and  the  polygons 
approximate  more  nearly  to  the  centrodes  for  the  given  motion;  and 
at  the  limit  these  polygons  reduce  to  a  pair  of  centrodes,  and  the 
temporary  centres  become  true  instant  centres. 

For  other  than  a  plane  motion  it  has  been  seen  (Art.  19)  that 
the  motion  must  be  referred  to  a  rotation  about  an  axis  instead  of  a 
centre.  The  locus  of  the  instant  axis  is  called  an  Axode. 

Centrodes  (or  axodes)  may  be  used  in  obtaining  a  motion  which 
is  too  complex  to  get  directly  by  the  usual  methods.  Several  desired 
positions  of  two  points  (a  and  b,  Fig.  80)  in  a  body  A,  relative  to 
the  points  m—n  of  a  body  M,  may 
be  laid  down,  and  the  centrodes  then 
derived  by  the  process  indicated  above. 
If  the  two  bodies^!  and  M  are  attached 
to  figures  having  these  centrodes  for 
contact  surfaces,  the  simple  rolling 
upon  each  other  of  these  surfaces  will 
produce  the  required  motion.  As  will  be  shown  in  a  later  chapter, 
in  treating  the  design  of  toothed  gearing,  it  is  possible  to  derive  a 
pair  of  gears  which  will  produce  a  motion  identical  with  the  rolling 
motion  of  these  centrodes  and  free  from  any  risk  of  slipping.  It  is 
mathematically  possible  to  secure  very  complicated  motions  by  the 
use  of  the  principles  given  in  this  article ;  but  there  are  many  prac- 
tical limitations  to  the  applications  of  such  a  process. 


CHAPTER  III. 

PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.     FRICTIONAL 

GEARING. 

44.  Nature  of  Rolling  Curves. — Since  the  condition  of  roll- 
ing action  is  that  the  contact  point  shall  always  lie  in  the  line  of 
centres,  the  contact  radii  must  both  coincide  in  direction  with  the 
line  of  centres  to  insure  pure  rolling,  and  as  the  contact  radii  lie 
in  one  straight  line  they  make  equal  angles  with  the  common 
tangent.  In  a  pair  of  curves  which  roll  upon  each  other 
{Figs.  81  or  82)  let  M  and  N  be  two  points,  one  on  each  curve, 


that  will  come  into  contact  when  the  radii  OM  and  OfN  are  in 
the  line  of  centre.  Then  the  radii  OM  and  O'N  must  make 
«qual  angles  with  the  tangents  to  the  curves  at  3/and  JV,  respec- 
tively; otherwise  these  radii  could  not  lie  in  one  straight  line  when 
the  two  tangents  coincide  at  contact  of  M  and  N.  Furthermore, 
the  arcs  PM  and  PJVmust  be  equal;  and  the  sum  of  the  radii  OM 
and  0'JVmust  equal  the  constant  distance  between  centres  0-0'; 
for  if  the  first  of  these  conditions  is  not  satisfied,  there  must  evi- 

78 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       79 

dently  be  some  sliding  action  between  the  curves;  if  the  second 
condition  is  not  fulfilled,  the  two  points  M  and  N  could  not  meet 
on  the  line  of  centres. 

Besides  pairs  of  circular  arcs,  in  which  the  condition  of 
pure  rolling  (but  not  that  of  positive  driving)  is  met,  there  are 
many  pairs  of  curves  that  satisfy  the  above  conditions.  Two  of 
these  forms  will  be  treated  in  detail,  and  a  general  practical  method 
will  be  given  for  deriving  a  curve  which  will  roll  with  a  given 
curve,  the  two  centres  being  fixed. 

45.  Rolling  Circles.  —  Figs.  64  and  65  show  pairs  of  tangent 
•circles  which  may  roll  upon  each  other,  for  the  contact  point  always 
lies  in  the  line  of  centres.  The  common  normal  passes  through 
both  centres  in  these  cases  so  motion  is  not  transmitted  positively; 
but  if  we  assume  that  there  is  no  slipping  between  these  curves 
the  linear  velocities  of  the  points  Pa  and  Pb  are  equal.  If  A  makes 
rii  revolutions,  and  B  makes  n^  revolutions,  per  unit  of  time  (call- 
ing the  radius  of  A  =  rl}  and  the  radius  ofB  =  r3),  the  linear  veloc- 
ity of  Pa  =  SflTiMi,  the  linear  velocity  of  Pb  =  %7rr9ny  and,  from 
the  assumption  of  no  sliding, 


(1) 


The  angular  velocity  of  A  is  GJi  =  Znni  ,  the  angular  of  B  is  G?,  = 
t  ,  but  from  equation  (1), 


r,        7tni       coi 

-i  =  -  -  =  —  =  a  constant:  .....     (2) 

rl      Barn,      GO, 

hence  the  angular  velocities  of  A  and  B  are  inversely  as  their  radii. 
This  familiar  relation  corresponds  with  the  relations  given  in  Art. 
34,  where  it  was  shown  that  in  any  case  of  rolling  curves  the  angu- 
lar velocity  ratio  is  inversely  as  the  lengths  of  the  contact  radii,  or 
inversely  as  the  perpendiculars  from  the  fixed  centres  to  the  common 
tangent.  This  relation  holds,  whether  the  angular  velocity  ratio 
is  constant  as  in  the  case  of  rolling  circles,  or  otherwise.  The  gen- 
-eral  theorem  of  Art.  29,  that  the  angular  velocity  ratio  is  inversely  as 


80  KINEMATICS  OF  MACHINERY. 

.the  perpendiculars  from  the  fixed  centres  to  the  common  normal, 
or  inversely  as  the  segments  into  which  the  line  of  centres  is  cut 
by  the  common  normal  is  not  applicable  to  the  special  case  of  tan- 
gent circles,  for  this  normal  coincides  with  the  line  of  centres,  and 
these  ratios  are  indeterminate.  Thus,  the  perpendiculars  from  0 

and  0'  upon  NN*  are  both  zero,  and  their  ratio  gives  —  =  ^ 

GO^        0 

The  common  point,  P,  of  Figs.  64  and  65  may  move  either  to 
the  right  or  the  left  along  the  common  tangent.  It  is  evident  from 
Fig.  64,  in  which  the  centres  lie  on  opposite  sides  of  the  path  of 
this  point,  that  the  rotations  of  A  and  B  are  opposite;  if  A  has  a 
right-hand,  negative,  or  clockwise  rotation,  B  has  a  left-handed, 
counter-clockwise,  or  positive  rotation;  or  when  the  circles  are  in 
external  contact  their  rotations  are  opposite.  On  the  other  hand, 
if  the  circles  are  in  internal  contact  (one  of  them  tangent  to  the 
concave  side  of  the  other,  as  in  Fig.  65)  the  rotations  are  both  in 
the  same  direction. 

All  the  statements  of  this  article  apply  to  circular  arcs  rotating 
about  their  centres  as  well  as  to  complete  circles;  except,  of  course, 
that  unless  the  curves  are  full  circles  the  action  is  limited,  and  must 
be  reciprocating. 

46.  Rolling  Ellipses. — Two  equal  ellipses,  each  rotating  about 
one  of  its  foci  as  a  fixed  centre,  with  a  distance  between  centres 
equal  to  the  common  major  axis,  will  roll  upon  each  other  without 
any  sliding  action. 

In  Fig.  82  two  such  ellipses  are  shown,  with  fixed  centres  at  the 
foci  0  and  0',  and  free  foci  at  Oi  and  OS.  00'  =  AB  =  A'  B'. 

It  is  a  property  of  the  ellipse  that  the  lines  drawn  from  any 
point  (M )  (Fig.  82)  to  the  foci  (0  and  Oi)  make  equal  angles  (a), 
with  the  tangent  (tm-tm\  and  also  that  OM  +  0,M '  =  AB. 

If  JV  is  a  point  similarly  located  in  an  equal  ellipse,  O'N  and 
and  0/JV^make  an  equal  angle,  a,  with  the  tangent  tn-tn.  Now 
the  two  ellipses  may  be  so  placed  together  that  M  and  N  will  co- 
incide at  the  contact  point,  when  the  tangents  tm-tm  and  tn-tn 
will  become  the  common  tangent,  and  OM  and  O'N  will  lie  in  one 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       81 

straight  line,  for  they  make  equal  angles  with  these  tangents.  If 
0  and  0'  are  made  the  fixed  centres  about  which  the  ellipses 
rotate  the  contact  point  lies  in  the  line  of  centres;  hence  the  action 
is  pure  rolling.  The  distance  00'  =  OM '-j-  O'N =  AB,  as  already 
stated.  Also,  0101'  =  A' B'  =  AB  =  00'. 

As  M  and  N  are  any  points  similarly  located  in  the  two  equal 
ellipses,  the  contact  point  will  always  be  in  the  line  of  centres  il 
the  conditions  as  to  these  centres  given  at  the  beginning  of  thi? 
article  be  observed. 

If  there  is.' no  sliding  between  the  two  ellipses  in  acting  through 
the  angles  POM'  and  PO'N',  respectively,  (Fig.  82),  the  arcs  PM' 
and  PN'  must  be  equal.  This  equality  can  be  shown  as  follows: 

OP  -f  Of  =  AB  =  A'B'  =  O'P  +  Oi'P,     .     .     (1) 

also  OP  +  O'P  =  AB  =  A'B'  =  0>P  +  0/P  ;    .     .     (2) 

.'.  OP  +  0*P  =  OP  +  O'P  =  O'P  +  0/P  =  OiP  +  OS  P.  .     (3) 

From  either  the  first  and  second,  or  the  third  and  fourth  mem- 
bers of  (3)  we  get: 

0>P  =  O'P, (4) 

from  which  it  is  seen  that  the  arcs  PB'  and  PA  are  equal. 

In  a  similar  way  it  can  be  shown  that  OiM'  =  0'N\9  and  that 
the  arcs  AM'  and  B'N'  are  equal;  therefore  the  arc  PM'  = 
AP  -  AM'  is  equal  to  the  arc  PN'  =  B'P  -  B'N'.  This 
demonstration  is  general  and  will  apply  to  any  pair  of  points 
which  can  meet  as  contact  points.  If  the  points  P  and  M  lie 
on  opposite  sides  of  AB,  and  P  and  N  lie  on  opposite  sides  of 
A'B',  the  values  of  PM  and  PN  become  PB  +  BM,  and  PA'  + 
A'Nj  respectively,  but  the  equality  of  the  arcs  is  maintained. 

The  driving  will  be  positive  in  the  direction  indicated,  until 
the  phase  shown  in  Fig.  83  is  reached,  when  the  normal  passes 
through  both  fixed  centres,  and  the  driver  might  continue  to  rotate 
without  imparting  further  motion  to  the  follower.  To  secure  con- 


82  KINEMATICS  OF  MACHINERY. 

tinuous  driving  for  the  half  revolution  succeeding  this  phase  it 
must  be  provided  for  otherwise  than  by  the  simple  contact  of  the 
two  ellipses.  It  has  been  shown  that  the  free  foci  Ol  and  O/,  are 
always  at  a  distance  apart  equal  to  the  major  axis,  A-B,  and  these 
foci  could  therefore  be  connected  by  a  link.  This  system  of  link- 
work  alone  would  transmit  motion  exactly  equivalent  to  that  of 


Fig.  83 


the  rolling  ellipses;  but  in  an  actual  mechanism  the  two  pieces  would 
have  to  be  at  the  ends  of  the  shafts  between  which  motion  is  to  be 
transmitted,  or  the  link  would  interfere  with  the  shafts.*  Another 
obstacle  to  such  a  link  connection,  as  a  substitute  for  the  rolling 
ellipses,  is  that  at  the  phase  shown  in  Fig.  83  (and  at  180°  from 
this  position)  the  linkwork  would  reach  a  "  dead-centre  "  position, 
when  it  would  not  be  effective  in  transmitting  motion. 

Teeth  may  be  placed  at  the  ends  of  the  elliptical  members  (as 
indicated  in  Fig.  83),  which  would  engage  near  the  dead-centre 
phases,  and  thus  carry  the  follower  past  this  critical  position.  If 
such  teeth  were  placed  around  the  entire  halves  of  the  ellipses 
which  are  in  contact  after  the  direct-contact  driving  ceases  to  be 
operative,  the  link  could  be  omitted,  and  the  necessity  of  placing 
the  ellipses  at  the  ends  of  the  shafts  thus  avoided.  Where  the 
action  is  to  continue  through  half  a  revolution,  or  more,  such 
teeth  are  usually  placed  entirely  around  the  peripheries  of  the 
ellipses,  and  the  result  is  a  pair  of  elliptical  gears  such  as  is  shown 
in  Fig.  84.  The  method  of  forming  such  teeth,  to  secure  the 
exact  equivalent  of  the  rolling  ellipses,  will  be  discussed  in  a  later 
chapter.  With  the  transmission  through  such  elliptical  members 

*  It  will  be  noted  that  the  pair  of  rolling  ellipses  correspond  to  the  centrodes 
of  such  a  system  of  links  as  that  just  suggested. 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       83 


as  have  just  been  discussed,  the  angular  velocity  ratio  is  inversely 
as  the  contact  radii  at  any  phase.     If  the  driver  has  a  uniform 


Fig. 84  A 

angular  velocity,  the  angular  velocity  of  the  follower  is  a  maximum 

in  the  phase  shown  in  Fig.  83,  when  — -  =  -^p  =  T^-     When  the 
driver  has  made  a  half  revolution  from  this  position,  the  angular 

velocity  of  the  follower  is   a  minimum,  and  — l  =  7^—. .     These 

G?a       OA 


84: 


KINEMATICS  OF  MACHINERY. 


extreme  ratios  are  reciprocals  of  each  other.  Of  course  the  driver 
and  follower  both  complete  the  half  rotations  from  these  two 
positions  (where  the  contact  radii  coincide  with  the  major  axes)  in 
equal  times.  If  it  is  required  to  connect  two  shafts  by  rolling 
ellipses  either  the  maximum  or  the  minimum  angular  velocity  of 
the  follower  may  be  taken  at  will,  but  one  of  these  being  deter- 
mined the  other  is  fixed — the  driver  being  supposed  to  have  a 
constant  angular  velocity. 

Suppose  it  is  required  to  construct  a  pair  of  rolling  ellipses  such 

that  the  maximum  value  of  — -  =  --.     Divide  the  distance  between 

oo,        I 

centres  00'  (Fig.  83)  into  such  segments  that  OP  :  O'P  ::  2  :  1. 
Lay  off"  PA  and  PB'  each  equal  to  00' ';  then  lay  off  PO,  and 
JB'O/  equal  to  PO'.  PA  and  PB'  are  the  major  axes  of  the  re- 
quired ellipses,  whose  foci  are  0  and  0, ,  and  0'  and  O/,  respec- 
tively; from  these  data  the  curves  can  be  constructed. 

Sectors  of  ellipses  can  be  used  for  transmitting  a  reciprocating 
motion  from  the  driver  to  the  follower.  In  this  case  the  angle 
through  which  one  of  the  members  turns,  and  both  the  maximum 
and  minimum  angular  velocity  ratios,  can  be  assumed;  but  the 
angle  through  which  the  other  member  rotates  is  not  then  subject 
to  control,  for  the  two  sectors  are  necessarily  alike.  Thus  (Fig.  85) 


Fig.  86 


the  centres  are  at  0  and  0',  and  it  is  required  to  construct  a  pair 
of  elliptical  sectors  such  that  an  angular  motion,  <*,  of  the  driver 
will  transmit  motion  to  the  follower  pivoted  at  0',  and  GOI  -f-  G?a  is 
to  have  for  extreme  values  O'P  -4-  OP,  and  O'P'  ~  OP'. 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       85 

•  Draw  a  Hue  from  0  making  the  angle  a  with  OP,  and  on  this 
line  lay  off  OM  =  OP'.  P  and  M  are  then  points  in  the  ellipse 
which  rotates  about  one  of  its  foci  at  0.  The  distance  from  /' 
to  the  free  focus  of  this  ellipse,  Ol ,  equals  the  major  axis  minus 
OP',  or  OtP  =  00'  —  OP  —  O'P.  With  this  length  as  a  radius 
and  P  as  a  centre  draw  an  arc,  ee.  Also,  the  distance  from  Ot  to 
M,  or  O^M,  =  00'  —  OM=  O'P'.  With  this  length  as  a  radius, 
and  a  centre  at  M,  draw  an  arc  ff.  The  intersection  of  the  two 
arcs  ee  and  ff  is  0,.  The  foci  being  located  and  the  major  axis 
known,  the  ellipse  can  be  drawn.  The  elliptical  arc,  PN9  of  the 
follower  is  equal  to  that  of  the  driver. 

The  constructions  just  outlined  apply  either  for  actual  rolling 
elliptical  members,  or  for  finding  the  "  pitch  curves "  for  toothed 
gears,  or  segmental  gears. 

Elliptical  gears  have  been  applied  in  many  cases  where  a  "  quick- 
return  "  action  is  required,  as  to  shaping-machines,  in  order  to  give 
a  quick  return  motion  to  the  tool  with  a  slower  stroke  during  the 
cutting.  They  have  also  been  used  to  actuate  the  slide-valve  in  a 
steam-stamp  used  for  crushing  rock,  where  it  is  desirable  to  admit 
the  steam  above  the  piston  throughout  nearly  the  entire  downward 
stroke  in  order  to  cause  a  more  effective  blow;  while  on  the  upward 
stroke  economy  demands  that  only  sufficient  steam  be  used  to  return 
the  stamp-shaft. 

47.  Rolling  Logarithmic  Spirals. — One  of  the  properties  of  the 
logarithmic  spiral  is  that  the  tangent  to  the  curve  makes  a  con- 
stant angle  with  the  radius  vector  at  all  points.  Owing  to  this 
property,  the  curve  is  also  called  the  equiangular  spiral. 

The  polar  equation  of  this  curve  is  0  =  logb  r,  in  which  b  is 
the  base  of  the  system  of  logarithms.  The  angle  made  with  the 
tangent  by  the  radii  vectores  is  different  for  different  values  of  5, 
but  it  is  constant  for  any  one  system  of  logarithms.* 

*  See  Fig.  86,  9  =  log^r.     Let  m  =  modulus  of  the  system  of  logarithms, 

dr   ,  rdB      rmdr 

,'.  dQ  =  m—  ;  but  tan  d>  =  -7—  = 5-  dr  =  m  =  the  modulus  of  the  sys- 

r  dr  r 

iem  of  logarithms;  .*.  0  =  tan-1  m  =  a  constant. 


86 


KINEMATICS  OF  MACHINERY. 


If  two  similar  logarithmic  spirals  are  placed  tangent  to  each 
other,  as  in  Fig.  87  or  88,  the  tangents  to  the  two  coincident  con- 
tact points  lie  in  the  same  line;  and  as  the  angles  made  by  these 
tangents  with  their  radii  vectores  are  equal,  these  radii  lie  in  a 
straight  line.  This  holds  for  all  tangent  positions  of  the  curves; 
hence  if  the  curves  turn  about  fixed  centres  at  their  foci,  the  con- 
tact point  always  lies  in  the  line  of  centres,  thus  meeting  the  re- 
quirement for  pure  rolling. 

The  sum  of  the  contact  radii  if  the  foci  are  on  opposite  sides  of 
the  contact  point,  and  their  difference  if  the  foci  are  on  the  same 
side  of  this  point,  is  a  constant  and  is  equal  to  the  distance  be- 
tween the  fixed  centres.  Thus,  in  Fig.  87,  OP  +  O'P  =  00'; 


o 


Fig.  88 


and,  if  r  and  s  are  two  points  which  may  become  coincident  con- 
tact points,  Or  +  O's  =  00'.     Also,  in  Fig.  88,  O'P  -OP  =00'; 
and,  if  r  and  s  are  two  points  which  may  become  coincident  con- 
tact points,  O's  -Or=  00'. 
In  Fig.  87: 

OP  +  O'P  =  Or  +  O's,    .'.  Or  -  OP  =  O'P  -  O's.  .     (1) 
In  Fig.  88: 

O'P  -  OP  =  O's  ~r  Or,    .'.  Or  -  OP  =  O's  -  O'P.  .     (2) 


Equations  (1)  and  (2)  show  that  in  either  external  or  internal 
contact   the  difference  between  two  contact  radii  of  one  of  the 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.      87 

spirals  equals  the  difference  between  the  corresponding  contact 
radii  of  the  other  spiral.  It  can  be  shown  that  any  two  arcs  ,  of 
similar  logarithmic  spirals  are  equal  in  length  when  the  difference 
of  the  radii  to  the  extremities  of  these  arcs  is  the  same.  Hence  in 
Figs.  87  and  88,  Pr  =  Ps,  as  it  should  for  pure  rolling.* 

A  single  pair  of  logarithmic  spirals  cannot  transmit  motion 
continuously  in  one  direction,  but  they  may  be  used  for  a  reciprocat- 
ing transmission  with  pure  rolling.  The  angular  motion  of  the  driver 
and  both  extreme  angular  velocity  ratios  may  be  assumed,  in  which 
case  the  angle  through  which  the  follower  moves  can  not  be  con- 
trolled. Thus,  in  Fig.  87,  the  driver  may  rotate  about  0  through 
the  angle  POr  =  a,  and  the  angular  velocity  ratio  varies  from 
O'P  -5-  OP  to  O's  -~  Or.  These  conditions  determine  the  points 
P  and  r  in  the  spiral  which  has  its  focus  at  0.  The  focus  0'  ',  the 
point  P,  and  the  length  of  a  second  radius  vector,  O's  =00'—  Or, 
are  also  fixed  for  the  second  spiral;  but  as  this  must  be  similar  to 
the  first  spiral,  the  angle  PO's  cannot  be  assigned  in  advance.  It 
is  possible  to  fix  the  angles  of  motion  of  both  driver  and  follower, 
but  with  these  conditions  only  one  angular  velocity  ratio  can  be 
taken  arbitrarily. 

48.  General  Case  of  Rolling  Curves.  —  A  general  method  will 
now  be  given  for  constructing  a  pair  of  curves  which  will  roll 
upon  each  other  in  turning  about  two  fixed  centres.  By  this 
method  the  angular  velocity  ratios  at  the  beginning  and  end  of 
any  angular  motion  of  one  member  may  be  assigned  ;  but  the  cor- 
responding angular  motion  of  the  other  member  cannot  be  pre- 
determined. Or,  if  one  of  the  curves  is  prescribed,  a  curve  can  be 
found  which  will  roll  upon  it.  The  method  gives  only  approxi- 

*   See  Fig.    86.      B  =  Iog6  r;  m  =  modulus,      (ds)*  =  (rdQ)'  -f  (dr}*\    but 


.-.  ds  =  V(m*  -f  l)dr,   .'.  8  =  v(m*  +  1)         dr  =  v(m«  +  l)(r,  -  r,);  hence 

«/r, 

the  length  of  tlie  arc  *  included  between  two  radii  vectores  of  the  same  differ- 
ence in  length  is  constant. 


88 


KINEMATICS  OF  MACHINERY. 


mate  results  inasmuch  as  it  does  not  absolutely  insure  theoretically 
perfect  rolling  between  the  points  located;  but  the  approximation 
can  be  carried  to  any  required  limit  by  locating  a  sufficient  number 
of  points. 

Suppose   the   distance   between   the   fixed  centres,  0   and   0', 
Fig.  89,  to  be  given,  and  that  it  is  required  to  construct  a  pair  of 

rolling  curves  such  that  the  angular  ve- 
locity ratio  of  B  to  A  shall  be  OP-f-  O'P, 
OP^O'P,,  OPt-t-0'Pt,  OP3-^0'P3, 
etc.,  when  the  lines  PO,  mvO,  m20,  w,0, 
etc.  respectively,  lie  in  the  line  of  cen- 
tres; these  last  lines  being  drawn  to  cor- 
respond with  required  angular  motions 
of  A. 

The  first  pair  of  radii  are  OP  for  A, 
and  O'P  for  B.  With  0  as  a  centre 
and  OP,  as  a  radius,  describe  the  arc 
Plml,  cutting  the  line  mvO,  then  draw 
With  0'  as  a  centre  and  O'P,  as  a  radius, 
now  take  a  radius  equal  to  Pm1;  with  P  as  a 
centre,  and  cut  the  arc  P^  at  nx;  and  connect  this  point  nx  with 
Pt.  It  is  evident  that  mt  and  n±  can  meet  in  the  line  of  centres 
when  A  has  turned  through  the  angle  mf)P  and  B  has  turned 
through  the  angle  n^O'P^.  Next  draw  an  arc  through  P2,  from 
centre  0,  cutting  the  line  m20  in  ra2,  and  connect  mt  and  m2. 
Also  draw  the  arc  P2n2  with  0'  as  a  centre  andO'P2  as  a  radius; 
now  with  a  radius  equal  to  ml  m2,  and  with  n1  as  a  centre,  cut  this 
last  arc  at  w2;  then  draw  the  line  nji^.  Proceed  in  a  similar  way 
with  the  points  P3,  P4,  etc.,  locating  the  points  ms,  w4,  etc.,  of  A ; 
and  w3,  nt,  etc.,  of  B.  It  will  be  seen  that  the  polygons  P-ml-mt 
.  .  .  mt,  and  P-nrn^  .  .  .  nt  may  act  together  with  a  rough  rolling 
action,  and  that  two  curves  can  be  passed  through  P-w^-ra,  .  .  . 
m6,  and  P~n^-n2  .  .  .  n6,  the  action  of  which  will  closely  approx- 
imate pure  rolling  if  the  points  located  are  sufficiently  close 
together;  that  is,  if  the  arcs  approximate  the  chords.  Evidently, 


a  line  from  P  to 
draw  the  arc  P 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       89 

if  the  outline  of  A  had  been  given,  the  curve  of  B  could  have 
been  derived  by  laying  off  the  lengths  Omlf  Om2,  etc.,  from  0 
upon  00',  and  then  proceeding  as  before  in  the  location  of  the 
points  of  the  outline  B. 

Fig.  90  shows  the  derivation  of  a  curve  B  to  roll  upon  the 
straight  line  which  rotates  about  0  as  a  centre  aud  constitutes  the 
acting  line  of  A.  The  construction  will  be  obvious  from  the  pre- 
ceding explanation  in  connection  with  Fig.  89. 

This  method  cannot  usually  be  applied  where  complete  rotation 
of  both  of  the  members  is  required;  for,  as  appears  from  the  con- 
structions given,  the  angular  motion  of  the  follower  for  a  given 


Fig.  90 


motion  of  the  driver  cannot  be  controlled ;  hence  it  is  not  certain, 
in  the  general  case,  that  a  complete  rotation  of  one  member  will 
correspond  to  a  complete  rotation  of  the  other.  But  with  con- 
tinuous action  in  one  direction,  when  one  member  has  turned 
through  360°  the  other  must  have  turned  through  an  angle  of  360°, 
or  else  some  exact  multiple  or  exact  divisor  of  360°.  This  require- 
ment does  not  apply  to  rolling  circles,  but  it  holds  for  all  other 
pairs  of  rolling  curves. 

49.  Lobed  Wheels. — It  has  been  seen  that  a  pair  of  equal  ellipses 
can  rotate  continuously  with  rolling  contact,  and  that  the  angular 
velocity  ratio  passes  through  one  maximum  and  one  minimum 
value  for  each  revolution.  It  is  sometimes  desirable  to  have  several 
maxima  and  minima  values  of  this  ratio  to  a  single  revolution,  and 


90 


KINEMATICS  OF  MACHINERY. 


a  class  of  rolling  mechanisms  called  Lobed  Wheels  may  then  be 
used. 

Fig.  91  shows  a  pair  of  these  wheels,  each  having  three  lobes. 
The  outlines  are  all  logarithmic  spirals. 

If  it  be  desired  to  have  an  unequal  number  of  lobes  on  the  two 


Fig.  91 


Fig.  92 


wheels  these  spirals  cannot  be  used;  but  curves  which  are  derived 
from  ellipses  permit  this  condition. 

Fig.  92  shows  a  set  of  three  such  wheels  in  series  which  roll 
perfectly;  there  is  a  one-lobed  wheel  acting  on  a  two-lobed  wheel,, 
and  this  latter  rolls  with  a  three-lobed  wheel.  These  figures  are 
drawn  from  MacCord's  Kinematics,  to  which  the  reader  is  referred 
for  a  full  treatment  of  Lobed  Wheels. 

In  all  of  these  wheels,  as  in  the  rolling  ellipses,  there  are 
periods  during  which  the  driving  is  not  positive;  but  these  outlines 
can  be  used  as  the  pitch  curves  for  toothed  wheels,  and  teeth  can 
be  formed  upon  these  curves  which  will  transmit  a  positive  motion, 
exactly  equivalent  to  that  of  the  pure  rolling  of  such  curves.  Ia 
these  derived  toothed  wheels  there  is  sliding  between  the  teeth 
themselves,  but  no  sliding  (if  the  teeth  are  properly  formed)  be- 
tween the  pitch  lines. 

50.  Rolling  Surfaces. — In  the  preceding  pages  plane  curves 
which  roll  upon  each  other  while  rotating  about  fixed  centres  have 
been  considered.  It  was  shown  in  Art.  10  that  the  plane  motion 
of  any  body  can  be  represented  completely  by  the  motion  of  a 
plane  figure ;  thus  these  plane  rolling  curves  may  represent  corre- 
sponding bodies  which  rotate  about  axes  through  the  fixed  centres- 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       91 

and  perpendicular  to  the  plane  of  motion.  When  two  or  more 
such  bodies  can  be  represented  by  figures  lying  in  the  same  plane, 
it  is  evident  that  the  axes  of  all  of  these  bodies  must  be  parallel. 
The  actual  contact  surfaces  of  such  bodies  are  generated  by  a  line 
which  travels  along  the  curved  outline,  always  remaining  parallel 
to  the  axes;  hence  these  surfaces  are  cylindrical.  The  actual 
bodies  corresponding  to  Figs.  64  and  65  are  figures  of  revolution  or 
right  cylinders  (see  Fig.  93);  while  the  bodies  corresponding  to 
Figs.  82  to  90  are  cylinders  only  in  the  general  sense.  Certain 
other  forms  may  roll  together  in  rotating  about  fixed  axes  which 
are  not  parallel,  when  the  motion  of  each  member  about  its  axis  is 
still  plane,  but  the  planes  of  motion  of  the  different  members  do 
not  coincide.  If  the  two  axes  intersect,  tangent  cones,  or  frusta 
(as  in  Fig.  95),  having  a  common  contact  element  and  a  common 
apex  at  the  intersection  of  the  axes,  may  act  together  with  pure 
rolling.  These  cones  are  not  necessarily  right  cones,  but  the  use  of 
cones  of  other  than  circular  transverse  sections  is  so  rare  that  only 
right  cones  will  be  treated  in  this  work. 

If  the  two  axes  are  not  in  one  plane  (i.e.,  if  they  are  neither 
parallel  nor  intersecting)  they  may  still  be  connected  by  two  mem- 
bers which  will  roll  upon  each  other,  with  contact  along  a  common 
rectilinear  element.  Fig.  101  shows  the  general  form  of  a  pair  of 
such  members;  they  are  called  Hyperboloids  of  Revolution.  The 
general  method  of  generating  these  latter  figures  and  the  nature  of 
the  action  will  form  the  subject  of  a  later  article,  in  which  it  will 
be  shown  that  there  is,  in  a  sense,  a  certain  departure  from  pure 
rolling  in  the  action;  however,  this  does  not  prohibit  the  use  of 
these  forms  as  pitch  surfaces  for  toothed  gears,  owing  to  the  pecul- 
iar character  of  the  sliding  component. 

51.  Rolling  Cylinders. — In  rolling  right  cylinders  the  angular 

velocities  are  inversely  as  the  radii;  or  — -  —  — •     Let  d  be  the  dis- 

GO,       rg 

tance  between  the  fixed  axes.  In  external  contact,  r1  -j-  r*  =  d ; 
and  in  internal  contact  rl  —  ra  =  d,  (in  this  expression  rl  is  taken 
as  the  radius  of  the  larger  cylinder,  inside  of  which  the  smaller  one 


92  KINEMATICS  OF  MACHINERY. 

rolls).  It  is  frequently  required  to  find  the  diameters  or  radii 
of  tangent  cylinders  which  will  connect  two  shafts  and  transmit 
motion  (when  there  is  no  slipping)  with  a  given  angular  velocity 
ratio.  This  ratio  is  the  same  as  the  ratio  of  the  revolutions  made 
in  a  given  time  by  the  two  cylinders,  and  in  practical  problems  it  is 
usually  stated  in  these  terms.  Thus,  one  shaft  is  to  make  nt  revo- 
lutions imparting  nt  revolutions  to  the  other  shaft,  per  unit  of 

fi>7  Al  A* 

time;   then  — -  =  —*=-?.      In   many   cases   the   required   radii, 

r,  and  ra,  can  be  found  by  inspection,  or  by  mental  calculation;  but 
it  may  be  convenient  to  use  the  following  expressions  if  nl  and 
w,  are  high  numbers  with  no  common  divisor. 

For  Cylinders  in  External  Contact :  r,  +  r2  =  a, .'.  TI  =  d  —  r2, 
and  r,  =  d  —  n. 


GO.       rt  GO  GO 

r±  —  —  ,     .*.  r.  =  r.  —  -  =  (d  —  r.)  —  *; 
<»,       r,'  '&?,  *  «V 


Similarly  :  r,  =  r,  ^  =  (d  -  r,)  -^  ; 


For  Cylinders  in  Internal  Contact  :  rl  —  ra  =  d  (r,  being  the 
radius  of  the  large  cylinder).     .'.  r,  =  d  -f  r,,  and  r,  =  rl  —  d. 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       93 


Similarly  : 


or 


or      r. 


d.    .   (4) 


(<»      \                     ,   G?,                                                                 ^,7                                            W, 
1 M  =  d— *;     .•.r.= ! — d',r2  = l- — 
09j          <»,'       •   '       (»,-  <»,              n,-  n, 


The  directions  of  the  rotations  of  the  two  members  are  oppo- 
site when  they  are  in  external  contact,  and  the  same  when  one  is 
tangent  to  the  concave  surface  of  the  other,  as  previously  pointed 
out. 

52.  Rolling  Right  Cones. — Two  right  cylinders,  combined  with 
two  right  cones,  are  shown  in  Fig.  94.  Each  cylinder  has  one 
base  in  common  with  that  of  one  of  the  cones,  hence  the  axis  of 
this  cylinder  and  cone  must  coincide.  The  bases  of  the  two  cones 
(and  of  the  corresponding  cylinders)  need  not  be  equal,  but  the 


Fig.  93 


Fig.  94 


Fig.  95 


slant  height  of  both  cones  is  the  same.  The  bases  of  the  two 
cones  have  a  common  tangent,  in  their  plane  (perpendicular  to 
the  paper),  passing  through  M.  Now  imagine  the  two  axes,  A  A 
and  BB,  to  rotate  in  their  common  plane,  about  this  tangent  to  the 
bases  through  M  as  an  axis  (or  hinge),  till  the  apex  a  meets  the 
apex  b  at  Q,  as  in  Fig.  95;  when  the  two  cones  become  tangent 
along  the  element  QM.  It  will  be  seen  that  the  two  base  circles 


94 


KINEMATICS  OF  MACHINERY. 


still  have  a  common  tangent  through  M  and  they  can  roll  upon 
each  other  in  the  new  position,  the  two  contact  points  having  equal 
velocities  along  their  common  tangent,  as  in  the  original  position. 
Any  other  corresponding  transverse  sections  of  the  cones,  equidis- 
tant from  Q  along  the  elements,  as  m-m'  and  m-m"  will  also  roll 
together;  or  the  two  cones  roll  upon  each  other  in  a  similar  manner 
to  the  original  rolling  of  the  cylinders. 

If  it  is  required  to  connect  two  given  intersecting  shafts  by 
rolling  cones,  so  that  their  rotations  per  unit  of  time  shall  be  in 
the  ratio  of  nl  to  n%9  it  is  only  necessary  to  construct  two  tangent 

right  cones  with  these  shafts  for  axes, 
and  with  a  common  contact  element 
lying  in  such  a  position  between  the 
axes  that  any  pair  of  transverse  sections 
which  roll  together  shall  have  radii  in 
LL_B^  the  inverse  ratio  of  the  required  angular 


Fig.  96 


motions.  If  A- A'  and  B-B',  Fig.  96, 
are  the  given  axes,  the  position  of  the 
contact  element  may  be  found  by  lay- 
ing off  from  Q,  on  these  shafts,  the 
distances  Qa  and  Qb,  directly  propor- 
tional to  the  required  numbers  of  rotations  of  these  shafts  ;  thus 
Qa  :  Qb  ::  nl  :  nv  On  Qa  and  Qb  form  a  parallelogram,  and  the 
diagonal  of  this  parallelogram,  Qc,  or  its  extension,  is  the  required 
common  contact  element. 

This  can  be  proved  as  follows  :  from  c  drop  perpendiculars  ce 
and  cf  upon  the  axes  A- A'  and  B-B';  the  angle  cbf=  eac  =  a  (sides 
parallel) ;  ce  =  ca  sin  at,  cf  =  cb  sin  a. 

.'.  ce :  cf  ::  ca  :  cb  ::  MM'  :  MM",  hence  the  cones  with  MM' 
and  MM"  as  the  diameters  of  the  bases,  will  roll  together  with  the 
required  angular  velocity.  The  frusta  used  for  this  transmission 
may  be  taken  from  any  part  of  the  two  cones,  giving  bases  greater 
or  less  than  those  indicated,  if  more  convenient. 

The  parallelogram  might  have  been  drawn  in  any  of  the  four 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       95 

angles  made  by  the  intersection  of  A- A'  and  B-B'  \  thus  if  the 
angle  B'QA  had  been  selected,  the  diagonal  Qc'  would  have  been 
located  for  the  contact  element,  and  two  such  frusta  as  those 
shown  with  M^M'  and  M^M"  as  bases  would  give  the  required 
angular  velocity  ratio.  Either  of  the  other  two  angles,  A'QB'  or 
A'QB,  might  have  been  taken  if  desired.  It  will  be  noticed  that 
the  cones  first  found  are  not  similar  to  those  obtained  in  the  second 
construction;  but  the  pairs  constructed  in  both  of  the  acute  angles 
are  similar,  as  are  the  pairs  in  both  of  the  obtuse  angles. 

If  the  driving-shaft  A- A'  rotates  as  indicated  by  the  arrows,  it 
will  be  seen  that  the  first  construction  (in  the  acute  angle)  imparts 
rotation  to  B-B'  in  one  direction  ;  while  the  second  construction 
(in  the  obtuse  angle)  causes  B-B'  to  rotate  in  the  opposite  direc- 
tion. The  choice  of  angle  for  the  location  of  the  contact  element 
is  governed  by  the  required  directions  of  the  rotations,  and  the 
locations  of  the  actual  shafts.  It  is  evident  that  one  of  the  ma- 


terial shafts,  but  not  both  of  them,  can  pass  through  Q.  Fig.  97 
shows  a  shaft  A-A',  from  which  four  shafts  (making  equal  angles 
with  A-A')  are  driven.  One  of  the  followers  on  either  side  of  A-A' 
is  rotated  in  one  direction  ;  while  the  other  followers  (one  on  each 
side  of  the  driver)  rotate  in  the  opposite  direction. 

It  may  happen,  as  in  Fig.  98,  that  one  wheel  cuts  through  the 


96 


KINEMATICS  OF  MACHINERY. 


axis  of  the  other  wheel.    The  shaft  can  then  be  led  off  only  ii:  the 

direction  indicated  by  the  full  lines  ; 
for  if  it  were  to  be  carried  through  Q, 
in  the  direction  of  the  dotted  lines, 
the  shaft  and  wheel  would  interfere. 
This  condition  can  only  occur  when 
the  contact  radius  is  located  in  the 
obtuse  angles.  The  acute-angle  con- 
struction is  to  be  preferred  as  avoiding 
this  difficulty  in  all  cases,  and  also 
because  it  gives  smaller  wheels  ;  but 
there  are  conditions  as  to  location  of  shafts  and  required  directional 
relation  of  rotation  which  may  make  the  other  construction  desir- 
able or  necessary.  The  conditions  of  the  problem  may  be  such  that 
the  contact  element  is  perpendicular  to  one  axis,  when  the  cone  on 
this  axis  is  of  the  special  form  (a  flat  disk)  shown  in  Fig.  99.  With 
somewhat  different  conditions,  one  of  the  rolling  surfaces  may  be 
the  concave  surface  of  a  cone,  as  shown  in  Fig.  100. 


Fig.  98 


a    Q 


Fig.  99  Fig.  100 

In  a  great  majority  of  the  cases  requiring  the  construction  of 
rolling  cones  on  intersecting  axes,  these  axes  are  at  right  angles  to 
each  other.  With  this  condition  the  pairs  of  cones  formed  in  any 
of  the  four  angles  (for  a  given  angular  velocity  ratio)  have  similar 
inclinations.  The  location  of  the  contact  radius  in  one  of  these 
angles,  and  the  selection  of  the  particular  angle  in  which  it  lies,  are 
determined  by  the  general  relations  previously  treated  in  this  article. 


PURE  ROLLING  IN  DIRECT-CONTACT  MECHANISMS.       97 


53.  Rolling  Hyperboloids. — If  one  right  line  revolves  about 
another  right  line  not  in  the  same  plane,  and  all  points  in  these 
lines  remain  at  constant  distances  apart,  the  revolving  line  gen- 
erates a  surface  called  the  hyperboloid  of  revolution.  This  is  a 
warped  surface,  the  elements  of  which  are  straight  lines  corre- 
sponding to  the  successive  positions  of  the  generating  line.  A 
meridian  plane  through  this  figure  cuts  the  surface  in  an  hyper- 
bola, and  it  is  evident  that  this  hyperbola  would  generate  a  sur- 


— B 


Fig.  101 


Fig.  101a 


face,  in  revolving  about  the  axis,  identical  with  that  generated 
by  the  straight  line;  hence  the  name  given  to  these  figures. 
Fig.  101  represents  a  pair  of  these  hyperboloids  of  revolution 
tangent  to  each  other  along  a  common  element  mm.  If  the  axes 
are  fixed  in  the  positions  corresponding  to  this  tangency,  it  is 
evident  that  the  two  surfaces  will  remain  tangent  as  the  two  figures 
rotate  about  their  axes;  for  each  is  symmetrical  about  its  axis 

Hyperboloids  of  revolution  can  be  placed  tangent  along  an 
element  only  when  the  radii  of  the  ft  gorge  circles  "  are  propor- 
tional to  the  tangents  of  the  angles  between  the  contact  element 


98  KINEMATICS  OF  MACHINERY. 

and  the  respective  axes;  i.e.,  when  P^  :PiBl  ::  tan  a  :  tan/J. 
This  is  shown  in  Fig.  10  la,  where  A  A  and  BB  are  the  two  axes 
and  mm  is  the  common  element.  AlBl  is  perpendicular  to  both 
axes,  and  PlAl  and  P±B±  are  the  respective  radii  of  the  gorge 
circles.  These  radii  are  normal  to  the  hyperboloids  and  intersect 
the  common  element,  mm,  which  is  therefore  perpendicular  to 
A1B1  at  Pi,  and  parallel  to  a  plane  through  A  A  perpendicular  to 
AJ$i.  B'B'  and  m'm'  are  the  projections  of  BB  and  mm  on  this 
plane ,  and  a  and  /?  are  equal  to  the  angles  between  mm  and  the 
respective  axes.  The  lines  cd  and  ce,  perpendicular  to  mm,  and 
intersecting  A  A  and  BB  aid  and  e,  respectively,  are  normals  to 
the  hyperboloids  at  c,  on  the  line  of  tangency.  Therefore  they 
lie  in  one  right  line,  de,  the  projection  of  which  on  the  plane  of 
AA  and  B'B'  is  de',  perpendicular  to  m'mf  at  c'.  P^c"  is  the  pro- 
jection of  P^c  on  the  plane  of  BB  and  A^B^  It  is  evident  that 
tan  a  c'd  cd  ccf  PiAl 
tan/?  =  cV  =  ce  =  c"e  =  PA' 

All  points  in  the  hyperboloid  which  rotates  about  A  A,  Fig. 
101,  must  move  in  planes  perpendicular  to  AA;  likewise,  all 
points  in  the  other  hyperboloid  move  in  planes  perpendicular  to 
BB,  and  as  the  two  axes  are  not  parallel,  two  contact  points 
can  not  have  identical  motions.  Thus  if  Va  is  the  velocity  of  a 
contact  point  in  the  former  figure,  Vb  is  the  simultaneous  velocity 
of  the  corresponding  point  in  the  latter  figure  when  they  roll 
together.  These  two  velocities  must  have  equal  components  per- 
pendicular to  the  contact  element,  but  their  components  along 
this  common  line  will  not  coincide.  This  is  the  characteristic 
of  the  action  of  these  bodies  referred  to  in  Art.  50,  and,  as  stated 
there,  it  does  not  affect  the  angular  velocity  ratio  of  the  two 
members,  for  this  relative  sliding  along  the  common  element  can 
not  transmit  motion,  nor  can  it  affect  the  component  of  Va  and 
Vb  perpendicular  to  the  common  element. 

The  angular  velocities  of  the  hyperboloids.  (Fig.  101)  when 
they  roll  together  are,  repectively,  cu1=Va-^PiAl  and  a>2  =  Vb  + 


FRICTIONAL  GEARING.  99 

P^j.     Va  =  V  +  cosa,   and  F6  =  F-^cos /?.     It  has  been   shown 
that  P^Al  :  PlBl  ::  tan  a  :  tan  /?.     Therefore 

ait       Va     PlBl  _  V  tan  £  cos  /?  _  sin  3 
a>2      Pi_Al     Vb        V  tan  a  cos  a      sin  a 

To  construct  a  pair  of  rolling  hyperboloids  to  transmit  motion 
between  two  shafts  with  a  given  angular  velocity  ratio : — project 
these  shafts  on  a  plane  parallel  to  both  of  them,  Fig.  101:  lay  off 
Pa  and  Pb  on  A  A  and  BB  proportional  to  the  required  revolutions; 
construct  the  parallelogram  P-a-c-b,  and  draw  PC;  this  locates  the 
projection  of  the  contact  element.  At  any  point  c  on  PC  erect  a 
perpendicular,  cutting  AA  and  BB  in  d  and  e  respectively.  Divide 
the  perpendicular  distance  (A^B^  between  A  A  and  BB,  at  Plt 
in  the  ratio  of  the  segments  cd  and  ce;  thenP1Al  and  P^  will 
be  the  radii  of  the  gorge  circles  of  the  required  hyperboloids. 

54.  Frictional  Gearing. — It  has  been  shown  that  two  axes, 
whether  parallel,  intersecting,  or  neither  parallel  nor  intersecting, 
may  be  provided  with  contact  members  the  surfaces  of  which  will 
roll  upon  each  other.  In  many  mechanisms  it  is  necessary  to 
maintain,  exactly,  a  prescribed  relation  between  the  motions  of  the 
members  throughout  the  entire  cycle  of  operations.  In  other 
instances  this  is  not  essential,  a  reasonable  departure  from  the 
precise  relative  motions  contemplated  being  permissible.  Thus 
in  cutting  a  screw-thread  in  a  lathe,  it  is  essential  that  the  relation 
between  the  rotation  of  the  spindle  and  the  translation  of  the  tool 
shall  be  strictly  constant,  and  the  positive  mechanism  (gears  and 
the  lead  screw)  insure  this  uniformity  of  action.  But  in  plane 
turning  the  feed  may  vary  somewhat  without  serious  results,  and 
the  belt-driven  rod-feed,  depending  upon  friction,  is  often  used, 
thus  saving  unnecessary  wear  of  the  screw.  It  sometimes  happens, 
as  in  machinery  subject  to  severe  shock,  that  a  positive  transmis- 
sion is  not  desired;  and  in  many  cases  this  is  not  an  absolute 
necessity.  When  a  limited  variation  of  the  motion  transmitted 
may  be  permitted,  and  the  two  shafts  to  be  connected  are  at  a 
considerable  distance  apart,  belting  or  rope  transmission  is  most 
often  employed.  Occasionally,  because  the  distance  between  the 
shafts  is  too  small  to  employ  these  methods  of  transmission 


100  KINEMATICS  OF  MACHINERY. 

advantageously,  or  for  other  reasons,  the  substitution  of  contact 
members  rolling  upon  each  other  is  convenient.  In  all  such  trans- 
missions having  circular  transverse  sections  the  action  is  purely 
frictional  throughout  the  revolution,  and  these  mechanisms  are 
classed  as  Frictional  Gearing. 

If  the  sections  are  non-circular  the  action  may  still  be  pure 
rolling,  as  shown  in  the  preceding  chapter;  but  the  driving  can- 
not be  positive  during  the  entire  rotation;  for  a  critical  phase  is 
reached  at  which  the  action  is  only  frictional,  and  beyond  this 
phase  driving  does  not  occur,  even  by  friction,  unless  other  ex- 
pedients (as  teeth)  are  introduced  (see  Fig.  83).  It  is  evident,  then, 
that  frictional  gears  must  have  circular  transverse  sections  in  order 
to  transmit  continuous  rotation. 

The  force  that  can  be  transmitted  through  frictional  gearing 
depends  upon  the  physical  character  of  the  surfaces  in  contact  and 
on  the  normal  pressure  between  the  two  surfaces.  Some  slipping 
or  "creeping"  almost  inevitably  occurs;  its  magnitude  depending 
upon  the  character  of  the  surfaces,  the  normal  pressure  between 
them  and  the  resistance  to  be  overcome. 

In  certain  applications  this  liability  to  slip  is  desirable  rather 
than  otherwise.  For  example,  in  hoisting,  where  it  is  not  essen- 
tial that  the  load  raised  shall  move  through  precisely  the  same 
distance  for  each  increment  of  motion  of  the  driver.  If  any  ob- 
struction to  motion  of  the  load  be  met,  the  slip  prevents  the  sud- 
den strain  (shock),  that  would  be  thrown  upon  the  entire  train  of 
mechanism  if  this  elasticity  (using  the  word  in  a  somewhat  popular 
sense)  were  absent.  If  a  car,  or  "  skip,"  in  being  hoisted  from  a 
mine  leaves  the  track,  meets  an  obstruction,  or  is  overwound,  the 
yielding  through  the  slipping  of  friction  gears  (or  of  belts)  lessens 
the  danger  of  breakage  over  that  encountered  with  a  positive  connec- 
tion. Furthermore,  these  friction  mechanisms  are  much  simpler 
in  design  and  construction,  and  quieter  in  running  than  toothed 
gears;  and,  owing  to  such  considerations,  the  employment  of  fric- 
tional gears,  or  "frictions,"  as  they  are  frequently  called  for  brev- 
ity, is  not  uncommon,  under  proper  circumstances. 


FRICTION AL  GEAKlffC.' " '' 


Frictional  gearing  is  important  in  itself,  and  the  study  of  it  also 
affords  a  good  basis  for  investigation  of  toothed  gearing. 

Kinematically,  any  of  the  figures  of  revolution  which  will  roll 
together,  as  pairs  of  right  cylinders,  right  cones,  or  hyperboloids  of 
revolution,  might  be  used  as  friction  gears;  but,  practically,  rolling 
cylinders  (Fig.  93),  and  the  disk  and  plate  ("  brush-wheel")  (Fig. 
102),  are  by  far  the  most  common  as  the  basis  of  such  gearing. 
Rolling  cones  are  also  used,  but  less  frequently. 

Two  cylinders  (Figs.  64,  65,  and  93)  may  be  used  to  transmit 
motion  and  energy,  up  to  the  limits  fixed  by  the  friction  at  the 
contact  element.  Supposing  no  slip  to  occur,  any  two  contact 
points  have  the  same  linear  velocity,  and  the  angular  velocities  of 
the  two  members,  A  and  B,  are  inversely  as  their  radii. 

If  it  is  required  to  impart  to  a  shaft  a  given  number  of  revolu- 
tions per  unit  of  time,  from  a  shaft  of  given  rotative  speed,  the 
distance  between  centres  being  also  determined;  the  required  radii 
can  be  found  by  the  expressions  of  Art.  51.  For  example,  d  =  48", 
nl  =  210  rev.  per  min.;  ny  =  270  rev.  per  min. 


The  solution  of  the  kinematic  part  of  this  problem  is  extremely 
simple. 

55,  Grooved  Frictions.  —  The  consideration  of  the  force  that  can 
be  transmitted  by  friction-gears  involves  the  normal  pressure  and 
the   coefficient   of  friction    between   the  contact 
surfaces.     This  consideration  often  modifies   the 
forms  of  the  members,  without  altering  the  kine- 
matic action;  and  in  many  cases  it  may  be  advan- 
tageous to  use  certain  derived  forms,  known    as 
grooved  frictions  or  "  V  "  frictions,   in   pluce   of 
the    fundamental    rolling    cylinders.       Fig.    103 
shows  a  pair  of  these  derived  forms  in  contact. 
It  will  be  seen  that  the  original,  or  ideal,  rolling 
cylinders  are  replaced  by  rolls  with  circumferen- 
tial grooves,  the  sections  of  which  (in  planes  pass-  "        Fig.  103 
ing  through  the  axis)  are  triangular,  or  more  usually,  trapezoidal. 


W2  KHJ-EM-AT1CS  OF  MACHINERY. 

The  actual  contact  surfaces  are  frusta  of  cones  of  equal  slant  and 
on  parallel  axes. 

In  order  to  discuss  the  action  of  these  grooved  rolls,  and  to 
understand  clearly  their  advantage  over  the  simple  rolling  cylin- 
ders, it  will  be  necessary  to  treat  briefly  the  action  of  the  forces  in- 
volved in  frictional  transmission. 

If  two  bodies  are  in  contact,  with  a  force  F  acting  in  the  direc- 
tion of  their  common  normal,  there  is  a  resistance  to  the  sliding  of 
one  body  upon  the  other,  and  this  resistance,  called  friction,  is  what 
makes  frictional  transmission  possible.  The  resistance  is  a  function 
of  this  normal  pressure  and  of  the  physical  character  of  the  sur- 
faces. If  the  surfaces  are  very  smooth,  the  resistance  under  any 
normal  pressure  becomes  comparatively  small.  If  rough,  the  pro- 
jecting particles  of  one  member  interlock  with  those  of  the  other 
and  the  friction  increases.  As  absolutely  perfect  surfaces  are  not 
attainable,  absolute  freedom  from  friction  (absence  of  this  resist- 
ance to  sliding)  is  impossible ;  and  the  greater  the  departure  from 
ideal  perfection  of  surface  (smoothness),  the  greater  is  the  friction 
between  any  given  pair  of  bodies.  The  friction  varies  inversely  as 
the  smoothness,  and  this  varies  both  with  the  nature  of  the 
materials  in  contact  and  with  the  degree  of  "finish."  In  every 
case  the  friction  is  greater  than  zero;  and  the  ratio  of  this  resist- 
ance,/, to  the  normal  force,  F,  is  called  the  coefficient  of  friction, 
/*.  This  coefficient  can  only  be  derived  from  experiment,  directly 
or  indirectly. 

Let  the  normal  pressure  between  the  surfaces  of  the  two  cylin- 
ders (Fig.  93)  be  represented  by  F.  According  to  Newton's  third 
law,  action  and  reaction  are  equal  and  opposite;  hence,  the  pressure 
of  A  towards  B  is  met  by  an  equal  and  opposite  pressure  of  B 
towards  A.  These  pressures  can  only  be  brought  to  bear  upon  the 
contact  surfaces  through  the  bearings  of  the  wheels  (neglecting 
weight),  and  the  action  and  reaction  at  the  bearings  are  equal ; 
therefore  a  pressure  F  must  be  exerted  by  the  bearings  upon  the 
axle  supported  by  them.  In  other  words,  the  pressure  between  the 
bearings  and  journals  equals  the  pressure  between  the  contact  sur- 
faces of  the  two  wheels.  As  the  bearings  themselves,  however  per- 


FR1CT10NAL  GEARING. 


103 


fectly  formed  and  lubricated,  are  not  frictionless,  the  normal  force, 
F,  necessary  to  transmit  energy  from  A  to  5,  involves  a  frictional 
action  at  the  bearings,  resulting  in  a  wasteful  resistance  to  be 
overcome,  and  also,  incidentally,  in  wear  of  these  parts.  It  is 
therefore  desirable  to  reduce  the  pressure  at  the  bearings  as  much  as 
possible;  but  the  friction  at  the  contact  surfaces  must  be  sufficient 
for  driving,  and  the  normal  pressure  at  these  surfaces  is  one  of  the 
elements  which  determine  this  friction.  It  is  in  order,  then,  to 
investigate  the  relation  between  the  bearing  pressure  and  the 
normal  pressure  at  the  contact  surfaces,  and  to  see  if  the  former 
can  be  reduced  without  undue  sacrifice  of  the  latter.  With  simple 
cylindrical  rolls  (Fig.  93)  the  total  bearing  pressure  for  each  wheel 
equals  the  normal  pressure,  F,  at  the  con- 
tact element.  In  the  case  of  "V"  frictions, 
however,  the  normal  pressure  between  the 
contact  surfaces  may  be  much  greater  than 
the  bearing  pressure.  This  can  be  shown 
in  connection  with  Fig.  104,  in  which  the 
wedge  of  A  is  inserted  in  the  corresponding 
groove  of  B.  The  common  normals  to  the 
contact  faces  of  A  and  B  through  the 
centres  of  the  faces  are  Pn  and  Pn'9  and 
the  normal  forces  between  these  faces  may 
be  taken  as  acting  in  the  lines  of  these  normals  (such  normal  forces 
are  really  the  resultants  of  systems  of  parallel  forces,  uniformly 
distributed  over  these  faces).  The  force  F,  acting  in  the  centre 
line  of  A  and  B  as  indicated,  passes  through  P,  and  it  can  be  re- 
solved into  components  along  Pn  and  Pn'  by  the  parallelogram  of 
forces.  These  components  are  represented  by  Fl  and  F^.  The 
effect  of  the  initial  force,  F,  is  equivalent  to  the  combined 
effect  of  its  components,  and  it  may  be  replaced  by  them; 
therefore,  the  effect  of  F  is  equivalent  to  the  normal  actions, 


Fig.  104 


F,  sin  d  +  *V  sin  0'  =  F  ;  and  Fl  cos  6  =  F,'  cos  6'.  If  6  =  6'  (the 
usual  condition),  Fj=F/,  and  the  total  normal  action,  F!+F/  = 
2Fi=F  -4-  sin0.  It  is  seen,  from  this  last  expression,  that 


104  KINEMATICS  OF  MACHINERY. 

the  normal  pressure  increases,  for  a  given  value  of  F,  as  6  becomes 
smaller.  When  6  =  90°,  the  total  normal  pressure  equals  F,  as  it 
should ;  for  in  this  case  the  groove  and  wedge  have  disappeared 
and  the  contact  surfaces  are  flat  and  perpendicular  to  the  line  of 
F.  For  any  value  of  0  less  than  90°,  the  total  normal  pressure  is 
greater  than  F. 

The  action  between  the  grooved  faces  of  the  "V"  frictions  is 
exactly  like  that  of  this  wedge.  The  normal  pressures  between 
the  sides  of  the  acting  ridges  and  the  grooves  correspond  to  the 
normal  pressures  in  the  wedge,  and  the  initial  force  F  equals  the 
radial  force  exerted  between  the  bearings  and  journals  of  the 
wheels.  It  follows  from  this  discussion  that  any  necessary  normal 
pressure  at  the  acting  contact  surfaces  of  the  "V"  frictions  can  be 
maintained  by  a  bearing  pressure  less  than  the  normal  pressure. 
Hence,  the  friction  loss  in  the  bearings  is  decreased  by  the  substi- 
tution of  the  ' '  V  "  frictions  for  the  fundamental  rolling  cylinders ; 
or,  to  state  it  somewhat  differently,  for  a  given  pressure  at  the  bear- 
ings, a  greater  resistance  can  be  overcome  at  the  rim  by '  'V  "  frictions 
than  by  cylindrical  rolls.  As  there  is  a  practical  limit  to  the  press- 
ure that  can  be  safely  carried  at  the  bearings,  and  as  excess  of 
bearing  pressure  means  waste  through  friction,  the  importance  of 
the  wedge-like  action  in  frictional  transmission  is  apparent. 

The  angle  between  the  sides  of  the  grooves  (2(9)  is  usually  from 
40°  to  50°.  Assuming  40°  as  this  angle,  V  =  20%  and  sin  0  =  0.342. 

F 

Then  2  Fl  —  -—  =  2.93  F ;  or  the  total  resulting  normal  pressure 
,o4/4 

is  nearly  three  times  the  force  at  the  bearing,  and  the  coefficient  of 
friction  and  bearing  pressure  remaining  the  same,  nearly  three 
times  as  great  a  resistance  can  be  overcome  with  these  grooved  rolls 
as  with  corresponding  true  cylindrical  rolls. 

Grooved  frictions  are  frequently  so  mounted  that  the  distance 
between  the  shafts  can  be  changed  somewhat.  This  makes  it 
possible  to  throw  the  wheels  out  of  gear,  so  that  the  follower  can 
be  stopped  without  checking  the  driver.  It  also  permits  control- 
ling the  bearing  pressure,  so  that  it  need  not  be  any  greater  than 
is  required  to  prevent  serious  slipping  at  the  driving  surfaces. 


FRICTION AL  GEARING.  105 

This  adjustment  also  affords  a  ready  means  of  taking  up  the  wear 
of  the  "V"  or  of  the  bearings,  so  that  good  contact  (without 
which  driving  is  impossible)  is  maintained. 

When  in  gear  (close  contact)  there  is,  as  usually  constructed,  a 
small  clearance  at  the  bottoms  of  the  grooves,  as  indicated  in  Fig. 
104.  If  this  were  not  provided,  the  edges  of  the  rings  might 
"bottom; "  that  is,  the  contact  might  be  entirely  or  mainly  at  the 
bottoms  of  the  grooves,  instead  of  at  the  inclined  sides.  Such  a 
condition  would  defeat  the  object  of  the  grooves,  and  to  render  it 
impossible,  even  after  considerable  wear  at  the  sides,  this  clearance 
is  provided. 

The  depth  of  the  grooves  of  either  wheel  is  the  difference 
between  the  radii  to  the  tops  and  bottoms  of  the  grooves  or 
rings.  This  distance  minus  the  clearance  at  the  bottoms  may  be 
called  the  ivorlcing  depth,  and  the  faces  of  the  "  Vs  "  above  the 
clearance  may  be  called  the  working  surfaces.  The  nominal  radius, 
or  pitch  radius,  of  a  grooved  friction  may  be  taken  as  the  mean 
radius  of  the  working  surface,  and  the  hypothetical  cylinder  corre- 
sponding to  this  radius  will  then  be  the  pitch  cylinder,  or  pitch 
surface. 

,  The  angular  velocity  of  two  V  friction  wheels,  when  in  full  con- 
tact and  working  properly,  may  be  taken,  for  most  practical  pur- 
poses, as  that  corresponding  to  the  rolling  together  of  the  pitch 
cylinders,  or,  inversely,  as  the  pitch  radii.  The  relative  sliding  or 
creeping  of  the  wheels  along  the  common  tangent  to  the  pitch  sur- 
faces may  usually  be  neglected  in  well-constructed  frictions;  for 
these  wheels  are  only  employed  where  some  variations  in  the  angu- 
lar velocity  ratio  is  admissible.  Assuming  that  no  sliding  of  this 
character  takes  place — that  is,  that  the  angular  velocities  of  the 
two  wheels  are  inversely  as  their  pitch  radii — there  is  nevertheless 
some  relative  motion  between  the  two  surfaces  when  they  are  in 
contact,  causing  a  grinding  action.  The  nature  of  this  action  may 
be  seen  in  connection  with  Fig.  105,  in  which  0  and  0'  are  the 
fixed  centres  of  A  and  B,  p  is  the  contact  point  at  the  pitch  circles, 
and  s  and  t  are  two  coincident  points  in  the  working  surfaces,  one 
on  each  side  of  p.  The  linear  velocity  of  p,  pv  is  assumed  to  be  the 


106  KINEMATICS  OF  MACHINERY. 

same  for  the  coincident  points  of  both  wheels  which  lie  at  p.  It  is 
evident  that  all  points  in  either  wheel  which  lie  outside  of  its  pitch 
circle  have  linear  velocities  greater  than  pv,  and  all  points  of  either 
wheel  lying  inside  of  its  pitch  circle  have  linear  velocities  less  than 
pv\  but  those  points  of  the  working  surface  of  one  wheel  which  are 
inside  of  the  pitch  curve  come  in  contact  with  points  of  the  other 
wheel  which  are  outside  of  its  pitch  circle;  consequently,  if  the 
points  at  the  pitch  circles  have  equal  linear  velocities,  all  contact 
points  not  in  these  circles  have  different  velocities,  and  there  must 
be  some  relative  motion  or  sliding  between  any  pair  of  such  points. 
Thus,  in  Fig.  105,  the  linear  velocity  of  s,  as  a  point  in  A,  is  sv' ; 

and,  as  a  point  in  B,  the  velocity  is 
sv"  ;  therefore  the  rate  of  sliding  of 
these  points  equals  sv"  —  sv'.    Simi- 
larly,   the   rate   of    sliding   at   t   is 
,s    ^''    XA      tvf  —  tv".     An  expression   for   the 
^X^X    greatest  sliding  is  derived  below.   Let 
^.^•'x      R  and  r  be  the  two  pitch  radii,  N 
~^^  'NB       and  n  be  the  numbers  of  revolutions 
per  unit  of  time  of  the  correspond- 
Fig.  105          ing  wiieeis^  and  ]i  the  working  depth 
&  of  the  grooves.    Then  the  velocity  of 


XL 


a  point  in  the  pitch  circle  of  either  wheel  is  ZnRN  =  ^nrn  . 
rn.  An  extreme  outer  point  of  the  working  surface  of  the  first 
wheel  has  a  radius  R  -+-  %h,  and  it  comes  in  contact  with  a  point  of 
the  other  wheel  having  a  radius  r  —  \h  ;  hence  the  sliding  at  these 
points  per  unit  of  time  equals 


snce         =  rn. 

By  taking  extreme  contact  points  on  the  other  side  of  the  pitch 
circles,  having  radii  R  —  %h  and  r  -j-  JA,  the  same  result  can  be 
reached  by  a  similar  process.  The  grinding  action  just  noted  tends 
to  wear  the  working  faces,  even  if  no  slipping  occurs  at  the  pitch 
circles.  Such  action  does  not  take  place  in  simple  cylinder  friction- 
rolls,  but  it  cannot  be  avoided  if  the  grooves  have  sensible  depth. 


FR1CTIONAL   GEARING. 


107 


The  rate  of  this  sliding  action  is  directly  proportional  to  h  ;  there- 
fore the  working  depth  should  be  as  small  as  practicable.  This 
dimension  is  limited  in  practice,  without  sacrifice  of  the  necessary 
total  contact  surface,  by  using  several  grooves,  side  by  side,  as  indi- 
cated  in  Fig.  103,  instead  of  fewer  and  deeper  ones. 

56.  Brush-wheels. — Fig.  102  shows  a  mechanism  sometimes 
used  where  it  is  desired  to  vary  the  angular  velocity  of  a  shaft 
which  is  driven  by  another  shaft  of  constant  angular  velocity. 
Suppose  the  plate  on  the  shaft  AA 
to  be  the  driver,  and  the  disk,  or 
"  brush  wheel,"  on  BB  to  be  the  fol- 
lower. A  long  keyway,  or  spline  (or 
its  equivalent),  permits  the  disk  to 
be  placed  at  different  positions  along 
a  line  parallel  to  a  diameter  of  the 
plate,  as  indicated  by  the  dotted 
locations.  The  disk  is  imagined  to 
be  of  no  sensible  thickness  ;  hence 
it  touches  the  plate  at  a  single  point, 
p.  This  point,  p,  is  at  a  distance  from  BB  equal  to  the  radius  of 
the  disk,  r'  ;  and  at  a  distance  from  the  axis  A  A  equal  to  r, 
which  may  vary  from  zero  to  R  (plus  or  minus).  Assuming  no 
slipping  at  p9  the  angular  velocity  ratio  of  AA  to  BB  is  r'  -r-  r 
(inversely  as  the  radii).  When  the  plane  of  the  disk  is  in  the  axis 
A  A,  the  velocity  of  p  is  zero,  hence  the  follower  is  at  rest.  If  the 
disk  is  carried  beyond  this  position  (to  the  opposite  side  of  AA), 
the  direction  of  the  rotation  of  the  follower  is  reversed.  This 
mechanism  is  not  well  adapted  for  heavy  forces;  but  is  very  con- 
venient in  many  cases  for  light  work,  as  in  feed  mechanism  and  for 
similar  purposes  requiring  considerable  change  in  the  rotative  speed 
of  a  follower,  or  reversal  of  direction  of  rotation.  The  disk  must 
have  sensible  thickness  in  practical  applications,  and  this  gives  rise 
to  a  grinding  action  somewhat  similar  to  that  mentioned  with  "V" 
frictions.  If  the  disk  is  a  cylinder,  with  contact  between  one  of  its 
elements  and  a  radius  of  the  plate,  it  is  evident  that  all  points  in 


Fig.  102     I A 


108  KINEMATICS  OF  MACHINERY. 

this  cylindrical  element  must  have  the  same  linear  velocity  (being 
points  in  a  body  at  the  same  distance  from  the  axis);  while  the  cor- 
responding contact  points  in  the  radius  of  the  plate  have  different 
linear  velocities  (being  at  different  distance  from  the  axis  of  this 
plate).  The  disk  should  therefore  be  as  thin  as  practicable,  and 
Its  edge  is  sometimes  rounded  to  approximate  the  point  contact  of 
the  ideal  disk.  The  plate  should  usually  be  the  driver  ;  for  if  this 
is  not  the  case,  when  the  disk  is  in  contact  with  the  centre  of  the 
plate,  the  latter  is  at  rest,  and  the  edge  of  the  disk  is  compelled  to 
slip  on  the  contact  surface. 

The  working  disk  is  often  made  of  leather,  wood,  or  other  yield- 
ing material,  held  between  metal  washers  of  slightly  smaller  diam- 
eter. This  construction  increases  the  adhesion,  and  makes  it  easier 
to  maintain  the  required  normal  pressure  at  the  contact  point  as 
slight  wear  takes  place.  In  other  friction  mechanisms  one  of  the 
members  is  frequently  made  of  a  non-metallic  substance  for  a 
similar  reason,  and  this  member  should  usually  be  the  driver;  for 
if  any  slip  occurs,  by  reason  of  the  resistance  being  greater  than 
the  friction  can  overcome,  the  tendency  is  to  wear  off  the  edge  of 
the  rotating  driver  evenly,  and  to  wear  a  depression,  or  notch,  in 
the  stationary  follower.  If  the  driver  is  made  of  the  softer  mate- 
rial the  more  irregular  and  objectionable  wear  of  the  follower  is 
thereby  reduced.  In  the  brush-wheel  mechanism  it  is  not  so  easy 
to  support  the  soft  face  on  the  driver  (the  plate),  and  there  is  not 
the  same  reason  for  doing  so;  because,  even  if  the  wear  were  all 
concentrated  on  this  face,  it  would  not  be  worn  off  evenly  all  over, 
for  the  follower  only  covers  a  small  portion  of  its  working  surface 
in  any  position.  When  the  follower  (disk)  is  at  the  centre  of  the 
plate  there  is  a  tendency  to  wear  a  small  flat  place  on  the  edge  of 
the  former.  This  may  be  avoided  in  many  cases  by  cutting  a  slight 
depression  at  the  centre  of  the  plate,  so  that  contact  does  not  take 
place  in  this  position  of  the  disk. 

57.  Cone  Friction. — Intersecting  axes  are  sometimes  connected 
by  rolling  conical  friction  wheels  similar  to  the  arrangements  indi- 
cated in  Figs.  96  to  100;  but  these  are  not  so  satisfactory  as  the 


FRICTIONAL   GEARING.  109 

frictions  on  parallel  axes,  as  it  is  more  difficult  to  adjust  the  posi- 
tions of  the  shafts  to  maintain  the  required  normal  pressure.  If 
the  force  to  be  transmitted  between  intersecting  axes  is  consider- 
able, it  may  be  better  to  use  positive  connections,  as  bevel-gears,  to 
connect  the  intersecting  shafts,  and  to  introduce  the  friction  ele- 
ment, if  necessary,  by  means  of  a  supplementary  shaft  parallel  to 
one  of  these. 

Two  cones,  as  shown  in  Fig.  106,  are  sometimes  used  to  connect 
parallel  shafts,  where  changes  in  the  angular  velocity  of  the  fol- 
lower are  required.  These  two  cones 
are  similar  in  inclination,  and  placed 
with  the  adjacent  elements  parallel, 
but  not  touching.  An  intermediate 
disk,  C  (or  its  equivalent),  capable  of 
being  moved  along  the  lengths  of  the  F'9-  l06 

cones,  is  in  contact  with  both  of  them.  Assuming  no  slipping  at 
either  contact,  the  linear  velocity  of  the  edge  of  this  disk  will  be 
that  of  the  part  of  the  driver  with  which  it  is  in  contact,  and  this 
same  linear  velocity  will  be  imparted  to  the  follower ;  hence  the 
linear  velocities  of  the  contact  points  of  the  two  cones  will  be  equal, 
and  the  angular  velocities  will  be  inversely  as  the  contact  radii  of 
the  cones  at  these  points.  If  the  disk  is  placed  nearer  the  large 
base  of  the  driver  it  acts  on  a  smaller  section  of  the  follower,  and 
the  angular  velocity  of  the  latter  is  correspondingly  increased. 

This  device,  or  modifications  of  it,  is  now  on  the  market,  for  use 
as  a  countershaft.  Provision  is  made  for  maintaining  proper  con- 
tact between  the  disk  and  the  cones.  In  this  case,  as  in  that  of  the 
brush-wheel,  the  disk  must  have  appreciable  thickness;  hence  its 
contact  element  engages  with  points  on  the  cones  which  must  have 
somewhat  different  linear  velocities,  and  a  corresponding  grinding 
action  occurs.  Similar  remarks  as  to  the  means  of  reducing  the 
practical  effect  of  this  action  apply  to  both  cases. 


CHAPTER  IV. 


OUTLINES    OF    GEAR-TEETH.     SYSTEMS  OF  TOOTH-GEARING. 

58.  Pitch  Surfaces. — It  has  been  shown  that  many  pairs  of 
bodies  (as  cylinders,  cones,  etc.)  may  transmit  motion  from  one  to 
the  other  with  pure  rolling,  while  these  bodies  rotate  about  axes 
fixed  in  the  proper  relative  positions;  but  that  the  action  of  the 
driver  upon  the  follower  is  not  continuously  positive. 

The  application  of  these  rolling  bodies  as  frictional  gearing  has 
already  been  treated.  There  are  many  cases,  however,  where  it  is 
desirable  to  secure  a  motion  equivalent  to  one  of  these  rolling 
actions,  but  where  it  is  absolutely  essential  that  no  practical  vari- 
ation from  this  prescribed  motion  shall  occur.  This  requirement 
is  frequently  met  by  using  the  surfaces  of  the  appropriate  rolling 
members  as  bases,  and  attaching  interlocking  teeth  to  them  to 
prevent  slipping.  These  rolling  surfaces,  when  so  used,  are  called 
pitch  surfaces ;  and  sections  of  them  perpendicular  to  their  axes 
are  called  pitch  lines,  or  pitch  curves. 

Toothed  gearing  may  he  classified  according  to  the  pitch  sur- 
faces, relation  of  the  axes,  and  character  of  the  tooth  elements  as 
follows :  * 


Class  of  Gearing. 

Relative  Position 
of  Axes. 

Pitch  Surfaces. 

Elements  of  Teeth. 

1 

Spur 

Parallel 

Cylinders 

Rectilinear 

2 

Bevel 

Intersecting 

Cones 

a 

3 

Skew 

In  different  planes 

Hyperboloids 

" 

4 

Twisted 

Any 

Either 

Helical 

5 

Screw 

In  different  planes 

Cylinders 

{( 

6 

Face 

Any 

None 

Circular 

*  MacCord's  Kinematics. 


110 


OUTLINES  OF  GEAR-TEETH. 


Ill 


The  action  of  these  various  classes  will  be  treated  in  detail  in 
later  articles. 

The  two  tangent  circles  of  Fig.  107,  representing  rolling  cylin- 
ders, may  have  their  circumferences  divided  up  into  arcs  of  equal 


Fig.  107 


length,  p  =  Pa  =  ab  =  be,  etc.,  =  Pa'  =  a'b'  =  b'c',  etc.  This 
length  of  arc,  p,  must  be  a  common  divisor  of  both  circumferences, 
and  the  numbers  of  divisions  on  the  two  circles  are  proportional 
to  their  circumferences,  diameters,  or  radii.  Let  the  radii  be  repre- 
sented by  r  and  r',  and  the  numbers  of  divisions  of  the  respective 
circles  be  called  t  and  t' ';  then  as 


p  p  t       * 

As  t  and  t'  are  directly  proportional  to  r  and  r' ,  it  follows  that  the 
angular  velocity  ratio  is  inversely  as  the  number  of  the  divisions 
of  the  two  circumferences.  It  is  to  be  noticed  that  a  and  a',  b  and 
b',  c  and  c',  etc.,  are  pairs  of  points  which  become  coincident  con- 
tact points  as  the  circles  roll  together. 

Now  if  we  bisect  the  arcs  Pa,  ab,  Pa',  a'b',  etc.,  and  place  pro- 
jections and  corresponding  notches  on  the  alternate  subdivisions, 
as  indicated  by  the  shaded  outlines  of  Fig.  107,  it  will  be  seen  that 
the  wheels  resemble,  somewhat,  the  familiar  toothed  gears.  The 
part  of  the  tooth  outside  of  the  pitch  circles  is  called  the  adden- 
dum or  point;  the  portion  inside  of  the  pitch  circle,  between  the 
spaces,  is  called  the  root.  The  acting  surface  of  the  point,  or  adden- 
dum, is  called  the  face,  and  the  acting  surface  of  the  root  is  called 


112  KINEMATICS  OF  MACHINERY. 

the  flank.  By  the  formation  of  such  teeth  the  pitch  circles  have 
lost  their  physical  identity,  but  they  are,  nevertheless,  important 
kinematically  as  the  basis  of  the  toothed  wheels.  The  distances 
Pa,  ab,  Pa' ,  etc.,  from  any  point  on  one  tooth  to  the  corresponding 
point  of  the  next  tooth  of  the  same  wheel,  measured  on  the  pitch 
curve,  is  called  the  circumferential  pitch,  circular  pitch,  or  simply 
the  pitch.  It  is  evident  that  the  pitch  must  be  the  same  for  both 
wheels. 

If  these  -wheels  are  "meshed"  (that  is,  placed  with  a  tooth  of 
one  in  a  space  of  the  other,  and  with  the  pitch  curves  tangent), 
as  shown  in  Fig.  107,  it  is  apparent  that  the  rotation  of  one 
of  them  will  cause  the  other  one  to  rotate,  and  that  the  transmission 
is  now  positive.  As  this  rotation  goes  on,  the  successive  pitch 
points  of  the  teeth  of  the  two  wheels  come  into  contact  on  the 
line  of  centres,  and  the  mean  angular  velocity  ratio  for  complete 
rotations,  or  for  angular  motions  of  the  wheels  measured  by  their 
pitch  arcs,  is  identical  with  that  due  to  the  pure  rolling  of  the 
pitch  circles.  This  might  be  sufficient  for  some  purposes  ;  but  we 
have,  as  yet,  no  assurance  that  this  angular  velocity  ratio  is  strictly 
constant  throughout  the  angular  movements  corresponding  to  the 
pitch  angles.  That  is,  the  mean  angular  velocity  ratio  during  such 
an  angular  motion  agrees  with  that  of  the  rolling  circles ;  but  at 
any  phase  intermediate  between  contact  at  two  pitch  points  the 
angular  velocity  ratio  may  be  either  greater  or  less  than  this  mean. 
It  is  imperative  in  many  cases,  and  desirable  for  smoothness  of 
action  and  quiet  running  in  nearly  all  cases,  that  the  angular 
velocity  ratio  be  constant  for  all  phases, 

59.  Conjugate  Gear-teeth. — The  condition  of  constant  angular 
velocity  ratio  in  direct  contact  is  that  the  common  normal  to  the 
acting  faces,  through  the  point  of  contact,  shall  always  cut  the  line 
of  centres  in  a  fixed  point;  hence  the  desired  constancy  of  this 
ratio  in  such  wheels  as  those  of  Fig.  107  demands  that  the  common 
normal  shall  always  pass  through  the  point  marked  P.  If  the 
teeth  are  of  such  form  that  this  condition  is  met,  the  motion  trans- 
mitted is  exactly  equivalent  to  the  rolling  of  the  pitch  circles,. 


OUTLINES  OF  GEAR-TEETH.  113 

otherwise  there  is    some    departure    from    the ,  required    relative 
motion. 

In  general,  the  form  of  the  teeth  of  one  wheel  may  be  taken 
quite  arbitrarily,  and  an  outline  can  be  found  for  the  teeth  of  the 
other  whe"el  which  will  give  the  required  angular  velocity  ratio  at 
all  phases;  but  this  statement  is  subject  to  practical  limitations. 
A  pair  of  teeth  which  work  together  properly  are  called  conjugate 
teeth. 

A  practical  mechanical  method  of  finding  a  conjugate  tooth 
outline,  when  both  pitch  curves  and  the  form  of  the  tooth  to  be 
mated  are  known,  will  be  explained  before  treating  the  formation  of 
teeth  geometrically.  This  method  is  applicable  when  it  is  required 
to  construct  a  wheel  to  mesh  with  an  existing  gear,  whether  the 
teeth  of  the  latter  have  lost  their  original  form  through  wear  or 
not ;  and  whether  the  pitch  curves  are  circles  or  not. 

Cut  out  two  segments  of  wood,  .1  and  B  (Fig.  108),  correspond- 
ing to  the  two  pitch  curves,  and  mount  them  on  centres  properly 
located.  Upon  the  segment  A,  repre- 
senting the  existing  gear,  attach,  in 
proper  position,  a  sheet  metal  templet 
corresponding  in  form  to  one  of  its 
teeth,  and  have  this  slightly  raised 
above  the  surface  of  the  wooden  seg- 
ment by  inserting  a  piece  of  thick  paper 
or  cardboard  between  them,  so  that  a  $  ./^  F; 

piece  of  drawing-paper  attached  to  the 
segment  B  can  pass  under  the  templet. 
Now  roll  the  segments,  without  slipping,  and  trace  the  outline  of 
the  templet  on  the  paper  attached  to  B  in  several  positions  quite 
close  together;  a  curve  tangent  to  all  of  these  tracings  of  the  tem- 
plet is  the  required  tooth  outline  for  B.  A  thin  strip  of  metal  be- 
tween the  edges  of  the  two  segments,  one  end  of  which  is  attached 
as  indicated  to  each  of  the  segments,  will  prevent  slipping  during 
the  operation.  It  is  evident  that  as  B  is  rolled  back  and  forth 
upon  A  the  outline  just  derived  on  B  will  always  be  tangent  to  the 


114  KINEMATICS  OF  MACHINERY. 

tooth  of  Ay  and  if  .B  is  provided  with  a  tooth  of  this  form  such  a 
tooth  in  acting  upon  the  given  tooth  of  A  will  transmit  motion 
identical  with  that  due  to  the  rolling  of  the  pitch  curves. 

The  method  just  explained  is  convenient  for  use  in  the  shop, 
and  it  suggests  a  corresponding  process  for  the  drafting-room. 

Draw  the  given  tooth  and  its  pitch  curve  upon  a  piece  of  heavy 
paper,  and  then  draw  the  pitch  curve  of  the  other  member  upon 
tracing-paper,  thin  celluloid,  or  other  transparent  material.  Place 
this  last  drawing  above  the  other,  with  proper  tangency  of  the 
pitch  curves,  and  trace  the  outline  of  the  given  tooth  upon  the 
tracing-paper;  roll  the  curves  through  a  small  arc,  being  careful  to 
avoid  slipping,  and  trace  the  tooth  outline  in  its  new  position;  re- 
peat this  operation  until  the  entire  arc  of  action  of  the  teeth  has 
been  covered,  and  then  draw  on  the  tracing-paper  a  curve  tangent 
to  all  of  the  tracings  of  the  given  tooth.  This  tangent  curve  is  the 
required  tooth  outline. 

From  what  has  preceded',  it  will  be  seen  that  two  cylinders  may 
be  provided  with  teeth  such  that  the  positive  motion  transmitted 
from  one  to  the  other  will  be  identical  with  that  of  the  two  cylin- 
ders when  rolling  upon  each  other  without  sliding.  This  applies 
to  cylinders  other  than  those  of  circular  cross-section  ;  for  the 
methods  of  finding  a  conjugate  tooth,  as  given  above,  apply  to  any 
pair  of  rolling  curves,  such  as  rolling  ellipses,  logarithmic  spirals, 
etc. 

60.  General  Method  of  Describing  Tooth  Ontlines. — The  general 
method  of  describing  gear-tooth  outlines  by  means  of  an  auxiliary 
rolling  curve,  or  generator,  will  be  developed  in  this  article. 

Suppose  A  and  B  (Fig.  109)  to  be  any  two  rolling  plane  figures 
upon  the  outlines  of  which  a  pair  of  gear-teeth  are  to  be  described. 
As  the  pitch  lines  are  rolling  curves  their  point  of  contact  is 
always  on  the  line  of  centres.  In  the  phase  shown  by  the  full  lines, 
the  angular  velocity  ratio  of  A  to  B  is  O'P  -f-  OP\  in  the  phase 
indicated  by  the  broken  lines,  this  ratio  O'P'  -f-  OP' ';  or  for 
any  phase  of  these  rolling  curves,  the  angular  velocities  of  the 
members  are  inversely  as  the  contact  radii.  If  a  pair  of  teeth  give 


OUTLINES  Of   GEAR-TEETH. 


115 


Fig.  109 


a  motion  identical  with  that  due  to  rolling  of  the  pitch  curves,  it  is 
evident  that  the  common  normal 
to  the  two  teeth  in  contact  must 
always  pass  through  the  point  on 
the  line  of  centres  at  which  the 
pitch  curves  are  tangent  to  each 
other;  for  these  teeth  are  exam- 
ples of  direct  contact  members,  in 
which  the  angular  velocities  are 
inversely  as  the  segments  into 
which  the  line  of  the  normal  cuts 
the  line  of  centres. 

If  such  a  figure  as  G  be  rolled 
upon  the  convex  side  of  the 
pitch  curve  of  A,  the  point  g  of  the  figure  G  will  trace  the  curve 
ga  on  the  plane  of  A.  Likewise,  by  the  rolling  of  G  on  the  con- 
cave pitch  curve  of  B,  the  point  g  will  generate  the  curve  gb  on 
the  plane  of  B. 

The  curve  G  is  the  generating  line  of  the  teeth  outlines,  and  it 
may  be  any  line  capable  of  rolling  on  the  convex  side  of  A  and  the 
concave  side  of  B.  The  point,  g,  in  this  generating  line  is  the 
describing  point  of  the  teeth.  Now  suppose  the  pitch  curves  and 
the  generating  line  to  be  in  the  positions  shown  by  the  broken 
lines,  with  the  generating  point  at  P'9  the  common  point  of  tan- 
gency  of  the  three  lines.  If  A  is  turned  to  the  right,  as  indicated 
by  the  phase  shown  in  full  lines,  B  will  turn  to  the  left  in  rolling 
upon  it,  and  G  can  be  rolled  upon  the  pitch  curves  so  that  it  remains 
tangent  to  both  of  them  at  their  contact  point  in  00'.  When  the 
pitch  lines  have  reached  such  a  position  as  is  shown  by  the  full  lines, 
G  will  lie  in  the  position  shown  by  the  full  line,  and  the  original 
contact  points  of  A,  B,  and  G  will  be  at  a,  b,  and  g,  respectively. 
The  arcs  Pa.  Pb,  and  Pg  must  be  equal,  as  the  action  has  been 
pure  rolling.  During  this  rotation  the  point  g  describes  a  curve 
upon  the  surface  of  A  (this  surface  being  supposed  to  rotate  with 
A  about  0)  such  as  ag,  as  noted  above;  g  has,  in'  a  similar  way, 


116  KINEMA  TICS  OF  MA  CHINEE  Y. 

generated  a  curve  bg  on  the  surface  of  B  (rotating  about  0'),  and, 
at  the  instant  under  consideration,  g  is  the  contact  point  common 
to  ag  and  bg.  Now  as  G  is  rolling  upon  the  pitch  curves  of  A  and 
B,  and  is  in  contact  with  them  at  P,  P  must  be  the  instant  centre 
of  G  relative  to  both  A  and  B'y  therefore  the  point  g  (a  point  in 
G)  is,  at  the  instant,  rotating  about  P,  and  its  motion  must 
be  in  the  line  gv,  perpendicular  to  gP.  As  the  point  g  is  generat- 
ing the  curves  ag,  and  bg,  the  common  tangent  of  these  curves 
must  coincide  with  the  line  of  motion  of  g  (gv),  and  gP,  perpendic- 
ular to  gv,  is,  therefore,  the  common  normal  to  ag  and  bg.  The 
curves  ag  and  bg  are  described  upon  the  surfaces  of  A  and  B9 
respectively;  and  it  is  evident  that  teeth  upon  these  members, 
having  the  outlines  ag  and  bg,  will  transmit  a  motion  exactly 
corresponding  to  that  of  the  rolling  pitch  lines;  because  their 
common  normal  passes  through  the  point  in  the  line  of  centres  at 
which  these  rolling  pitch  curves  are  tangent  to  each  other. 

The  reasoning  of  the  foregoing  discussion  is  perfectly  general. 
It  applies  to  any  phase,  if  the  condition  that  the  three  curves  roll 
together  with  a  common  contact  point  is  met  at  every  instant  of 
the  action;  hence  the  curves  derived  by  this  construction  fully 
satisfy  the  kinematic  requirements  of  tooth  outlines. 

The  discussion  immediately  following  will  be  confined  to  wheels 
having  circles  (or  circular  arcs)  for  pitch  lines. 

61.  Usual  Systems  of  Gearing. — There  are  a  great  many  curves 
that  can  be  used  as  generating  lines  for  the  outlines  of  gear-teeth, 
but  only  two  are  commonly  used,  viz.,  circles  and  right  lines. 

The  curve  traced  by  a  point  in  a  circle  as  it  rolls  upon  the  con- 
vex side  of  another  circle  is  called  an  epicycloid;  if  it  rolls  upon 
the  concave  side  of  another  circle,  the  curve  traced  is  &liypocydoid\ 
and  if  it  rolls  along  a  straight  line  a  cycloid  is  described.  When  a 
right  line  rolls  upon  a  circle  any  point  in  this  line  traces  a  curve 
called  an  involute. 

The  common  systems  of  gearing  in  which  the  teeth  are  generated 
by  circular  or  rectilinear  describing  lines  are  called,  respectively, 
the  Epicycloidal  System  and  the  Involute  System. 


OUTLINES  OF  GEAR-TEETH.  117 

62.  Epicycloidal  Gearing. — Fig.  110,  A  and  B  are  two  pitch 
circles,  with  centres  at  0  and  0',  and  tangent  at  the  point  P. 
The  generator  or  describing  circle, 
G,  has  its  centre  at  o,  on  the  line  t 

of  centres  00' .     If  these  circles  \ 

all  turn  about  their  respective 
centres  (rolling , upon  each  other), 
the  paths  of  these  points,  a,  6, 
and  a,  which  originally  coincided 
at  P,  will  be  along  the  arcs  Pa, 
P6,  and  Pg.  Since  there  is  roll-  / 

ing  contact  Pa,  P6,  and  Pg  are  i  pjg.  no 

all  of  equal  length.  During  this 
motion  the  point  g  will  generate  an  epicycloid  by  rolling  on  the 
outside  of  A,  and  a  hypocycloid  by  rolling  on  the  inside  of  B. 
At  any  instant  these  two  curves  will  be  in  contact  at  g,  in  the 
circumference  of  the  describing  circle.  As  the  instant  centre  of 
the  generator,  relative  to  either  of  the  pitch  circles,  is  always  at  P, 
g  moves  perpendicular  to  Pa,  and  Pg  is  normal  to  both  curves  at 
their  point  of  contact.  This  normal  always  passes  through  P, 
hence  the  angular  velocity  ratio  is  constant. 

The  curves  just  discussed  are  suitable  for  the  outlines  of  gear- 
teeth,  and  if  the  driver,  A,  has  teeth  with  epicycloidal  faces,  and 
the  follower,  B,  has  teeth  with  hypocycloidal  -flanks,  generated  by 
the  same  circle,  G,  the  action  would  begin  at  the  pitch  point,  P 
and  continue  through  a  period  depending  upon  the  length  of  the 
teeth. 

It  is  evident  that  a  generator  G'  could  be  made  to  describe 
flanks  for  A  and  faces  for  B,  as  shown  by  the  curves  a'g',  and 
b'g',  respectively,  which  would  satisfy  the  conditions  of  constant 
velocity  ratio,  and  that  the  action  of  this  pair  of  curves  is  entirely 
independent  of  the  first  pair  ;  hence  G  and  G'  may  be  any  two 
circles. 

At  the  sides  of  the  figure  are  shown  complete  teeth  of  A  and 
B,  the  outlines  of  which  correspond  to  the  curves  traced  by  the 


118  KINEMATICS  OF  MACHINERY. 

describing  circles  G  and  G' '.  The  faces  of  A  and  the  flanks  of  B 
are  the  epicycloid  and  hypocycloid  generated  by  (7,  and  are 
identical  in  form  with  the  curves  ag  and  bg,  respectively.  The 
faces  of  B  and.  flanks  of  A  are  of  the  forms  generated  by  G',  as 
shown  by  Vg*  and  a'g',  respectively.  The  teeth  are  symmetrical; 
therefore  either  side  may  be  the  acting  side,  and  either  wheel  may 
drive. 

If  the  common  pitch,  p,  is  an  exact  divisor  of  both  circumfer- 
ences; if  the  lengths  of  the  teeth  are  such  that  at  least  one  pair 
shall  always  be  in  contact;  and  if  the  spaces  are  deep  enough  to 

allow  the  points  to  clear  in  passing  the 
centre  line,  these  wheels  will  meet  all 
essential  requirements. 

63.  Action  of  Epicycloidal  Gear 
Teeth.— In  Fig.  Ill  the  tooth  outlines, 
at  the  left  are  just  coming  into  contact, 
and  those  at  the  right  are  just  quitting, 
contact.  The  angle  aOa'  through  which 
gear  A  turns  while  one  of  its  teeth  is 
in  contact  with  a  tooth  of  B,  is  called 
Fig.  in  '  the  angle  of  action  of  A.  The  angle 

aOP,  passed  through  while  the  contact  point  is  approaching  the 
pitch  point  is  the  angle  of  approach  of  A,  and  angle  POa'  passed 
through  while  it  is  receding  from  the  pitch  point  is  the  angle  of 
recess.  The  angles  of  action,  approach,  and  recess  of  B  are  bO'b' ', 
bO'P,  and  PO'b' ',  respectively.  The  path  of  the  point  of  contact 
during  approach  is  along  the  arc  gP  of  the  describing  circle  G, 
and  during  recess  it  is  along  the  arc  Pg'  of  the  describing  circle  G'. 
It  has  been  shown  (Art.  62)  that  the  arcs  Pa,  Pb,  and  Pg  are 
of  equal  length.  The  arcs  Pa',  Pb' ,  and  Pg'  are  also  of  equal 
length.  Therefore  the  arcs  of  action,  aPa'  and  bPb' ',  subtended 
by  the  respective  angles  of  action  are  of  equal  length,  and  this; 
length  is  equal  to  that  of  the  path  of  the  point  of  contact,  gPg' '. 
When  the  point  of  contact  is  at  P ,  the  teeth  have  pure  roll- 
ing action;  at  all  other  times  the  action  is  mixed  sliding  and 


OUTLINES   OF   GEAR-TEETH.  119 

rolling  (Art.  36) .  The  rate  of  sliding  is  greatest  when  the  point 
of  contact  is  farthest  from  the  pitch  point,  and  it  decreases  to 
zero  at  the  pitch  point.  Since  this  sliding  causes  friction 
it  is  desirable  to  reduce  it  to  a  minimum.  Decreasing  the 
length  of  teeth  lessens  the  angle  of  action  and  the  length  of 
the  path  of  contact,  and  therefore  reduces  the  sliding.  For  con- 
tinuous action,  one  pair  of  teeth  must  come  into  contact  before 
the  preceding  pair  quits  contact  ;  therefore,  the  angle  of  action 
cannot  be  less  than  the  angle  subtended  by  the  pitch  arc;  or 
the  arc  of  action  aPa'  (or  bPb')  must  at  least  equal  the  dis- 
tance between  similar  points  (on  the  pitch  line)  of  two  adjacent 
teeth  of  either  wheel.  This  condition  fixes  the  minimum  length 
of  the  teeth.  If  the  given  pitch  of  the  two  wheels  (Fig.  Ill) 
is  aar  —  W,  this  determines  the  minimum  arc  of  action.  This  arc 
may  be  distributed  in  any  way  between  the  approach  and  recess 
arcs,  though  these  are  commonly  nearly  equal.  Lay  off  Pa  =  Pb 
=  Pg,  and  Pa'  =  Pb'  =  Pgr  equal  to  the  desired  arcs  of  approach 
and  recess,  respectively;  then  g  and  g'  are  the  extreme  points  in  the 
faces  of  B  and  A,  respectively ;  or  circles  drawn  with  the  radii  O'g  and 
Og'  are  the  boundaries  of  the  teeth  of  the  two  wheels.  The  strength 
of  the  teeth  depends  upon  their  thickness,  and  the  pitch  is  ordi- 
narily twice  the  thickness  of  the  teeth  at  the  pitch  circle,  or  slightly 
greater  to  allow  clearance  at  the  sides,  which  is  called  "backlash;" 
thus  the  pitch  is  a  function  of  the  force  to  be  transmitted.  As  has 
been  shown,  the  arc  of  action  must  at  least  equal  the  pitch;  it  is 
often  made  great  enough  to  insure  that  two  teeth  shall  always  be 
in  contact ;  or  that  as  one  pair  is  in  contact  at  the  centre  Ihie,  the 
preceding  pair  shall  be  quitting  contact,  and  the  succeeding  pair 
shall  be  beginning  contact.  This  requires  an  arc  of  action  tqual  to 
twice  the  pitch  arc,  and  correspondingly  longer  teeth,  for  i  given 
pitch. 

The  force  acting  between  the  teeth  is  transmitted  in  the  direc- 
tion of  the  common  normal  (neglecting  the  effect  of  friction),  or  in 
a  line  through  P  and  the  contact  point  of  the  teeth.  This  contact 
point  always  lies  in  the  describing  circle  G  during  approach,  and 


120  KINEMATICS  OF  MACHINERY. 

in  Gf  during  recess ;  hence  it  appears  that  the  force  transmitted  is 
more  oblique  as  the  contact  point  is  removed  from  P.  The  effect 
of  this  obliquity  is  to  increase  the  pressure  between  the  teeth  and 
at  the  bearings,  with  a  corresponding  increase  in  the  energy  wasted 
through  friction.  The  friction  due  to  the  sliding  action  of  the 
teeth  tends  to  increase  the  obliquity  of  the  pressure  between  the 
teeth  during  approach  and  to  decrease  it  during  recess  by  the 
amount  of  the  angle  of  friction.  Consequently  the  action  during 
recess  is  smoother  than  it  is  during  approach.  For  this  reason 
gears  are  sometimes  made  in  which  the  action  is  confined  to  the 
angle  of  recess,  in  which  case  the  driving  gear  has  faces  only,  and 
the  driven  gear  has  flanks  only.  Wheels  of  smaller  pitch  have 
shorter  teeth,  other  things  being  equal,  and  their  action  is  smoother 
under  the  ordinary  conditions  because  the  contact  point  is  always 
nearer  the  line  of  centres,  where  the  rate  of  sliding  of  the  teeth 
upon  each  other  is  less. 

It  will  be  seen  that,  for  a  given  pitch,  the  length  of  teeth  re- 
quired for  a  given  arc  of  action  is  less  as  the  describing  circles  used 
are  larger  in  diameter. 

64.  Determination  of  Describing  Circles. — During  contact  the 
faces  of  the  teeth  of  A  act  only  upon  the  flanks  of  the  teeth  of  B', 
similarly,  the  faces  of  B  act  only  on  the  flanks  of  A',  hence  the 
form  of  the  faces  of  one  wheel  does  not  affect  that  of  its  own  flanks 
nor  of  the  faces  of  the  mating  wheel.  There  is  no  necessary  fixed 
relation  between  the  two  describing  circles  G  and  G' . 

If  the  describing  circle  has  a  diameter  equal  to  the  radius  (£  the 
diameter)  of  the  pitch  circle  within  which  it  rolls  in  tracing  a  hypo- 
cycloid,  this  special  hypocycloid  is  a  right  line  passing  through  the 
centre  of  the  latter  circle,  or  a  diameter  of  it.  Hence  if  the  describ- 
ing circles,  G  and  G'  (Fig.  110),  have  diameters  equal  to  the  radii  of 
B  and  J,  respectively,  both  wheels  will  have  radial  flanks ;  but 
these  will  operate  properly  in  conjunction  with  the  corresponding 
epicycloidal  faces.  The  faces  would  not,  in  this  case  have  the  forms 
shown  *n  Fig.  110,  as  the  faces  of  one  wheel  and  the  flanks  of  the 
other  one  must  be  derived  from  equal  describing  circles.  The 


OUTLINES   OF  GEAR-TEETH.  121 

radial  flank  forms  are  simple  in  construction  and  describing  circles 
are  som  itimes  used  for  a  pair  of  gears  which  will  give  such  teeth. 
If  the  describing  circle  has  a  diameter  less  than  the  radius  of  the 
pitch  circle  within  which  it  rolls  in  tracing  the  hypocycloid,  the 
flanks  lie  outside  of  radii  through  the  pitch  point  ;  while  if  the 
diameter  of  the  describing  circle  is  greater  than  the  radius  of  this 
pitch  circle,  the  hypocycloidal  flanks  lie  inside  of  the  radii  to  the 
pitch  points.  The  first  of  the  forms  gives  spreading  flanks  which 
are  much  stronger  than  the  converging  or  undercut  flanks  of  the 
latter  form.  The  radial  flank  is  intermediate  between  these  forms 
in  strength.  Except  in  small  gears  (frequently  called  pinions)  for 
light  work,  undercut  flanks  are  seldom  used;  the  radial  flank  usu- 
ally being  the  weakest  form  allowed.  While  it  is  desirable  for 
strength  of  the  teeth  to  have  spreading  flanks,  and  therefore  to  use 
a  small  describing  circle,  large  describing  circles  give  teeth  which 
act  upon  each  other  with  less  obliquity. 

In  exceptional  cases,  when  a  single  pair  of  gears  are  to  work 
together,  it  may  be  good  practice  to  choose  the  largest  pair  of  de- 
scribing circles  which  will  give  the  necessary  strength  of  flanks, 
and  flanks  of  a  comparatively  weak  form  may  be  used  by  giving  a 
small  excess  to  the  pitch  (thickness  of  teeth).  In  such  cases  of 
single  pairs  of  gears,  for  reasons  already  given,  radial  flanks  will 
sometimes  be  used  for  both  wheels.  In  making  a  set  of  patterns 
(or  cutters  for  cut  gears),  however,  it  is  desirable  on  the  score  of 
economy  to  provide  for  the  working  of  any  wheel  of  the  set  with 
any  other  wheel  of  the  same  pitch.  If  this  is  possible  the  set  is  said 
to  be  interchangeable.  Suppose  that  in  the  two  gears,  A  and  B  (Fig. 
110),  the  faces  of  the  former  and  the  flanks  of  the  latter  are  gener- 
ated by  a  describing  circle  G,  and  that  the  faces  of  B  and  the  flanks 
of  A  are  generated  by  another  circle  G'.  It  has  been  shown  that 
these  two  wheels  will  work  together.  A  third  wheel,  C,  of  the  same 
pitch,  can  not  work  properly  with  both  A  and  B ;  for  if  the  faces  of 
C  are  generated  by  G,  and  its  flanks  are  generated  by  G',  it  may 
engage  with  #;  but  it  can  not  act  correctly  with  A,  for  the  faces  of 
A  and  the  flanks  of  C  are  not  generated  by  the  same  circle ;  neither 


122 


KINEMATICS  OF  MACHINERY. 


are  the  flanks  of  A  and  the  faces  of  (7,  and  the  conditions  of  con- 
stant velocity  ratio  are  not  met  by  this  construction. 

If  G  =  Gf,  C  won  Id  work  correctly  with  either  A  or  B,  or 
with  any  other  wheel  of  the  same  pitch,  the  faces  and  flanks  of 
which  are  epicycloids  and  hypocycloids  generated  on  its  pitch  line 
})jG=  G'.  We  may  then  state  that:  The  conditions  necessary 
in  an  Interchangeable  Set  of  Gears  are  that  all  of  the  ivheels  of  the 
set  shall  have  the  same  pitch,  and  that  the  teeth  of  all  of  them  shall 
have  faces  and  flanks  generated  by  equal  describing  circles. 

It  is  common  to  assume  that  the  smallest  wheel  that  will  prob- 
ably be  required  will  be  a  pinion  of  either  12  or  15  teeth,  and  to 
take  a  describing  circle  which  will  give  radial  flanks  to  such  a 
pinion;  that  is,  a  describing  circle  with  a  diameter  half  that  of  the 
pitch  circle  of  this  smallest  pinion.  If  t  is  the  number  of  teeth  in 
the  smallest  pinion,  its  pitch-circle  radius,  or  the  diameter  of  the 

describing  circle,  =  ——. 

65.  Annular  Wheels. — Fig.  65  shows  two  rolling  circles,  one  of 
which  is  tangent  to  the  concave  side  of  the  other.  The  correspond- 
ing rolling  cylinders  may  be  used  as  pitch  surfaces  of  gears.  The 
larger  of  these  is  called  an  annular  gear. 


Fig.  112 


The  method  of  generating  the  teeth  of  such  gears  is  indicated 
in  Fig.  112,  and  it  is  similar  to  that  explained  for  external  gears> 


OUTLINES   OF  GEAR-TEETH.  123 

except  that  the  faces  of  B  and  flanks  of  A  (generated  by  G)  are 
both  epicycloids,  and  the  faces  of  A  and  the  flanks  of  B  (generated 
by  G')  are  loth  hypocycloids. 

66.  Rack  and  Pinion. — If  one  pitch  line  is  a  right  line  (a  circle 
of  infinite  radius),  as  shown  in  Fig.  113,  teeth  may  be  formed  by 
a  method  similar  to  that  given  for  the  more  general  case  of  spur 
gearing.  Such  .a  gear  is  called  a  rack,  and  the  wheel  which  meshes 
with  it  is  usually  called  a  pinion.  The  faces  and  flanks  of  the 
rack  are  loth  cycloids ;  and  they  are  alike  in  an  interchangeable 
set  of  gears,  where  but  one  describing  circle  is  used.  In  such  a 
set,  any  wheel  will  engage  properly  with  the  rack.  The  construc- 
tion of  teeth  for  a  rack  and  pinion  is  shown  at  the  left  of  Fig.  1 13  ; 
and  at  the  right,  the  complete  teeth  are  shown  in  the  acting 
positions.  Of  course  the  rack  is  necessarily  of  limited  length,  . 


Fig.  113 


and  the  motion  transmitted  between  a  rack  and  pinion  must  be 
reciprocating. 

67.  Pin  Gearing.— If  the  describing  circle  equals  one  of  the 
pitch  circles,  the  hypocycloid  in  this  pitch  circle  becomes  a  mere 
point;  and  this  point  acting  on  an  epicycloid  generated  on  the 
other  wheel  by  this  same  describing  circle  will  transmit  a  motion 
identical  with  the  rolling  of  the  two  pitch  circles.  Fig.  114 
shows  such  a  point  in  B  acting  on  the  epicycloidal  faces  of  A.  In 
an  actual  gear  a  pin  of  sensible  diameter  must  be  used,  and  Fig. 
114  shows  such  a  pin,  and  dotted  line  curves  parallel  to  the 


124 


KINEMATICS  OF  MACHINERY. 


Fig.  114 


original  epicycloid  of  A   and  at  a  distance  from  this  epicycloid 
equal  to  the  radius  of  the  pin.     This  pin  and  the  dotted  outline 

will  transmit  the  same  motion  as  that 
due  to  the  point  and  epicycloid. 

With  the  point  and  the  epicycloid 
the  angle  of  action  is  entirely  on  one 
side  of  the  line  of  centres,  and  the  pin 
gear  should  always  be  the  follower, 
in  order  that  the  action  shall  take 
place  during  recess  rather  than  ap- 
proach. With  a  pin  of  sensible  diam- 
eter the  action  begins  at  a  distance, 
practically  equal  to  the  radius  of  the 
pin,  before  the  line  of  centres  is  reached,  and  there  is  consequently 
also  an  angle  of  approach.  The  derived  curve  of  the  driver  gives 
shorter  teeth  than  the  full  epicycloids,  and  the  height  of  the 
•driver's  teeth,  above  the  pitch  line,  is  therefore  diminished, 
thus  decreasing  the  angle  of  recess.  These  gears  were  formerly 
much  used,  when  teeth  were  commonly  made  of  wood,  as  the  pin 
I'orm  is  easily  constructed;  but  this  class  of  gearing  is  now  used 
but  little,  except  for  light  gearing,  such  as  clockwork,  etc. 

68.  Involute  Teeth. — When  a  right  line  rolls  on  the  circum- 
ference of  a  circle  any  point  of  the  line  traces  an  involute  of  the 
circle.  It  is  a  property  of  this  curve  that  the  normal  at  any  point 
is  tangent  to  the  base  circle.  In  Fig.  115,  if  the  right  line  EE' , 
tangent  to  the  base  circles  aa  and  66,  has  rolling  contact  with 
these  circles  as  they  rotate  about  the  fixed  centres  0  and  0',  re- 
spectively, any  point  g  of  EE'  traces  an  involute  of  each  base 
circle.*  In  every  phase  of  this  operation  these  two  involutes  are 
tangent  to  each  other  at  the  position  of  g,  and  the  line  EE'  is 

*  The  right  line  EE  may  be  considered  as  the  tangent  portion  of  a  flexible 
band  which  wraps  upon  one  base  circle  and  unwraps  from  the  other,  as  they 
rotate.  The  point  g  in  this  band  generates  the  two  involutes  upon  the  rotat- 
ing planes  of  the  respective  circles. 


OUTLINES   OF   GEAR-TEETH.  125 

normal  to  both  curves  at  this  point.  This  common  normal 
always  cuts  the  line  of  centres,  00',  in  a  fixed  point,  P.  If  these 
involute  curves  are  used  as  the  out- 
lines of  teeth  for  two  gears  A  and  B, 
turning  about  the  fixed  centres  0  and 
0'  respectively,  a  constant  angular 
velocity  ratio,  equivalent  to  pure  roll- 
ing contact  between  two  pitch  circles 
tangent  to  each  other  at  P,  will  be  A 
maintained  as  long  as  the  involutes 
are  in  contact.  When  A  turns  in  a 
clockwise  direction,  the  first  contact 
between  the  tooth  curves  occurs  when  Fjg< 

the  tracing  point  is  at  E,  and  contact  continues  as  g  moves  along 
the  line  EE'  until  E'  is  reached.  For  continuous  action  with  teeth 
of  involute  outline  the  pitch  angles  must  not  exceed  the  angles 
through  which  the  respective  gears  have  turned  during  this  time. 

The  line  EE'  is  the  locus  of  the  point  of  contact.  The  angle 
between  EE'  and  the  common  tangent  to  the  pitch  circle  is  the 
angle  of  obliquity.  Standard  gear  cutters  are  so  made  that  the 
sine  of  the  angle  of  obliquity  is  0.25,  which  corresponds  to  an 
angle  of  14£°. 

It  is  an  important  property  of  involute  tooth  outlines  that 
the  distance  between  the  centres  of  rotation  may  be  changed 
without  affecting  the  velocity  ratio.  Whatever  the  distance 
between  the  centres  of  the  base  circles  of  the  two  involutes,  the 
common  normal  to  both  curves,  in  any  position  of  tangency,  is 
always  tangent  to  both  base  circles,  and  divides  the  line  of 
centres  into  segments  which  are  proportional  to  the  radii  of  the 
respective  base  circles.  Since  the  angular  velocity  ratio  is  in- 
versely proportional  to  the  ratio  of  these  segments,  it  is  independent 
of  the  distance  between  centres.*  This  property  is  peculiar  to 

*  It  will  be  noted  that  the  mathematical  pitch  circles  vary  in  diameter 
with  such  adjustment  of  the  centre  distance,  and  the  pitch  circles  of  involute 
gears  have  not  the  same  physical  significance  as  in  epicycloidal  gears. 


126  KINEMATICS  OF  MACHINERY. 

the  involute  system,  and  is  exceedingly  valuable,  especially  in 
in  gears  connecting  roll -trains,  in  change  gears,  etc.,  where  exact 
spacing  of  the  centres  can  not  be  maintained.  When  a  pair  of 
rolls  connected  by  involute  gears  becomes  worn,  or  when  adjust- 
ment is  necessary  for  passing  material  of  different  thickness 
between  them,  the  centre  distance  may  be  changed  considerably 
(if  sufficient  initial  backlash  has  been  provided  between  the  teeth) 
without  affecting  the  angular  velocity  ratio.  The  limits  of  allow- 
able adjustment  of  the  centre  distance  are  reached  when  it 
becomes  so  great  that  as  one  pair  of  teeth  are  engaging  the  pre- 
ceeding  pair  are  quitting  contact,  and  when  it  is  so  small  that 
the  backlash  is  reduced  to  zero,  on  account  of  the  greater  thick- 
ness of  the  teeth  inside  the  original  pitch  line. 

The  angles  through  which  the  gears  may  be  turned  while  a 
a  pair  of  involute  tooth  outlines  are  in  contact  depend  on  the 
angle  of  obliquity.  When  the  angle  of  obliquity  is  zero,  the  base 
circles  coincide  with  the  pitch  circles,  and  the  involutes  are 
both  entirely  outside  the  pitch  circles,  and  can  not  come  into 
contact  except  when  passing  the  pitch  point.  The  angle  of 
action  is  zero.  This  represents  a  special  (though  impossible) 
case  of  epicycloidal  gearing  in  which  the  diameter  of  the  describ- 
ing circles  is  increased  to  infinity.  As  the  angle  of  obliquity 
is  increased  the  angle  of  action  also  increases. 

The  length  of  teeth  necessary  for  this  action  is  indicated  by 
the  circles  through  E  and  Ef  with  centres  at  0'  and  0  respectively- 
Since  the  distance  between  either  of  these  addendum  circles 
and  the  corresponding  pitch  circle  always  exceeds  that  between 
the  pitch  circle  and  the  base  circle  of  the  mating  gear,  it  is  neces- 
sary to  extend  the  tooth  spaces  inside  the  base  circles  to  accommo- 
date the  ends  of  the  teeth.  These  extensions  of  the  tooth  out- 
lines are  usually  radial  lines  tangent  to  the  involute  curves  at  the 
base  line  and  to  fillets  at  the  roots  of  the  teeth.  Involute  teeth 
are  sometimes  called  teeth  of  single  curvature,  as  there  ia  not  a 
reversal  of  curvature  at  the  pitch  line. 


OUTLINES  OF  GEAR-TEETH. 


127 


The  tooth  outlines  of  an  involute  rack  are  composed  of  straight 
lines  perpendicular  to  the  locus  of  the  point  of  contact.  Fig. 
lloa  shows  an  involute  rack  and  pinion  in  mesh. 


69.  Interference  in  Involute  Teeth. — The  length  of  standard 
gear  teeth  is  determined  by  their  pitch,  rather  than  by  the  con- 
siderations stated  in  the  preceding  article.  When  one  gear  has 
a  small  number  of  teeth  of  standard  proportions  the  ends  of  the 
teeth  of  the  mating  gear  extend  beyond  the  point  of  tangency 
of  the  common  normal  and  the  base  circle.  This  is  shown  in 
Fig.  116,  which  illustrates  a  pair  of  teeth  in  contact  at  the  point 
of  tangency  between  the  base  circle  of  the  smaller  gear  B  and 
the  common  normal.  The  part  of  the  tooth  outline  of  B  inside 
the  base  circle  is  a  radial  line.  It  is  evident  that  any  further 
turning  of  A  toward  the  right  will  result  in  contact  with  the 
radial  part  of  the  outline  of  B,  and  the  angular  velocity  ratio 
will  not  be  constant.  This  contact  of  the  teeth  inside  the  base 
circle  is  called  interference.  To  avoid  interference  the  flanks 
of  the  teeth  of  B  may  be  hollowed  out  or  the  points  of 
the  teeth  of  A  may  be  cut  away.  The  latter  is  the  usual 
remedy. 


128 


KINEMATICS  OF  MACHINERY. 


If  the  portion  of  the  face  which  comes  into  contact  with  the 
radial  flank  of  the  mating  tooth  is  given  the  form  of  an 
epicycloidal  arc  generated  on  the  pitch  circle  by  a  describing 
circle  of  half  the  pitch  diameter  of  the  mating  wheel,  the 
action  will  be  correct,  for  the  radial  flank  is  equivalent  to  a 
hypocycloidal  flank  formed  by  this  same  describing  circle. 


Fig.  116 

This  is  strictly  correct  only  if  the  given  centre  distance  is 
maintained. 

The  least  number  of  teeth  that  an  involute  gear  of  14^°  obliq- 
uity may  have  and  mesh  without  interference  with  an  equal 
gear  is  23;  the  least  number  in  a  gear  that  will  mesh  with  a  rack 
without  interference  is  32. 

70.  Comparison  of  the  Systems — Since  involute  teeth  trans- 
mit constant  angular  velocity  ratio  when  the  centre  distance 
varies,  exact  setting  is  not  so  necessary,  and  wear  of  the 
bearings  does  not  disturb  the  action  as  it  does  in  the  epi- 
cycloidal system. 


OUTLINES   OF  GEAR-TEETH.  129 

The  line  of  action  is  always  in  the  same  direction,  and 
the  force  acting  between  the  teeth  is  nearly  constant  in  the 
involute  system;  while  the  acting  force  is  variable  both  in 
direction  and  magnitude  in  the  epicycloidal  system.  The  former 
teeth  wear  more  evenly  as  a  consequence.  The  mean  thrust 
on  the  bearings  is  slightly  greater,  but  more  uniform,  with  in- 
volute teeth. 

All  involute  teeth  have  the  same  generator;  hence  the 
gears  are  interchangeable  if  of  the  same  pitch.  Epicycloidal 
teeth  are  better  for  low-numbered  pinions,  but  otherwise  have 
no  great  advantage  and  many  disadvantages.  They  are,  how- 
ever, quite  commonly  used  for  gears  having  cast  teeth;  while 
the  involute  system  has  largely  supplanted  the  epicycloidal 
system  for  cut  gears. 

71.  Clearance  and  Backlash.     The  term  backlash  has  already 
been  explained  as  the  clearance  at  the  sjdes  of  the  teeth;  it  is 
equal  to  the  width  of  a  space  minus  the  thickness  of  a  tooth,  both 
measured  on  the  pitch  line. 

The  backlash  provides  for  any  irregularity  in  the  form  or  spac- 
ing of  the  teeth.  It  may  be  very  small  in  accurate  cut  gears;  but 
must  be  larger  in  cast  gears. 

The  spaces  are  always  made  deeper  than  is  required  to  allow  the 
points  of  the  teeth  to  pass;  this  allowance  is  called  bottom  clear- 
ance, or  simply  clearance ;  it  also  provides  a  lodging-place  for  a 
moderate  quantity  of  dirt  or  other  foreign  substance  which  may 
get  between  the  teeth. 

72.  Pitch  of  Gear  Teeth.— The  action  of  the  teeth  is  smooth- 
est   when    the    contact    point    is    near    the    line    of    centres; 
hence  a  large  number  of  small  teeth  gives  more  uniform  action 
than  fewer  and  larger  teeth.     The  teeth  must  be  thick  enough 
to  sustain  the  load,  however,  and  the  pitch  is  determined  by  this 
consideration.     The    formula    W^spfy,    known    as    the    Lewis 
formula,  is  in  general  use  for  determining  the  load  that  may 
be  carried  by  the  teeth  of  gears.     In  this  formula  ir  =  the  total 


130  KINEMATICS  OF  MACHINERY. 

load  transmitted  by  the  teeth,  in  pounds;  s  =  the  safe  working 
stress  allowed  in  the  teeth,  in  pounds  per  square  inch;  p  =  the 
circular  pitch,  and/=  the  width  of  face,  both  in  inches;  ?/= a  factor 
depending  on  the  form  of  the  teeth.  For  standard  epicycloidal 

and  14£°  involute  teeth,  y==  0.124—       — ;  where  n  =  ihe  number 

of  teeth  in  the  gear.  The  pitch  of  such  teeth  necessary  to  sustain 
a  working  load,  Wlt  per  inch  of  width  of  face,  for  a  gear  of  given 
diameter,  D,  as  determined  from  the  above  formula  is: 


.     .     .     (1) 


The  relation  between  the  circular  pitch,  the  number  of  teeth, 
and  the  pitch  diameter  of  a  gear  is  expressed  by  the  equation, 
pn=nD,  from  which  p  =  r:D  +  n,  D  =  pn+-K,  and  n  =  7zD  +  p. 
When  either  the  pitch  or  the  diameter  is  taken  of  a  convenient 
size,  the  other  must  be  expressed  as  a  decimal  value.  Thus  the 
pitch  diameter  of  a  gear  having  36  teeth  of  1J"  circular  pitch 

36  X 1  5 
is  -=17.19",  while  if  the  diameter  is  taken  as  18"  the 

7C 

pitch=  =1.571".     It  is  much  more  convenient  to  express 

ob 

the  tooth  spacing  in  terms  of  the  number  of  teeth  per  inch  of 
diameter  of  the  gear.  This  ratio  is  called  the  diametral  pitch, 
for  which  the  symbol  p'  is  used.  The  relation  between  the  dia- 
metral pitch,  the  number  of  teeth,  and  the  pitch  diameter  of  a 
gear  is  expressed  by  the  equation  pf  =n  -f-  D,  from  which  D=n  +  p', 
and  n=Dp'.  For  a  gear  18"  in  diameter  and  having  36  teeth, 
p'  -36-7-18=2. 

The  relation  between  diametral  and  circular  pitch  is  found  by 
combining  the  equations  for  the  value  of  D.     D= 
.'.  p  =  7r-t-pf,  and  p'  =  x-t-p. 


OUTLINES  OF  GEAR-TEETH.  131 

Expressed  in  terms  of  the  diametral  pitch  equation  (1)  becomes 

p^-l  (0.194  +  ^0.037-2.15^').  i     ' 

73.    Proportions    of   Gear    Teeth. —  The    dimensions   of   all 
other  parts  of  gear  teeth  are  usually  expressed  as  functions  of 


Fig.  117 


the  pitch.     Fig.  117  will  serve  to  explain  the  terms  and  symbols 
used  for  these  parts. 

p  =  circular  pitch  (measured  on  the  pitch  line). 
t= thickness  of  tooth  (measured  on  the  pitch  line). 
s^ width  of  space  (measured  on  the  pitch  line). 
a  =  addendum  =•  length  of  tooth  outside  the  pitch  line. 
d= working  depth  of  tooth  =  2a. 
c— clearance. 

h  =  whole  depth  of  tooth  =  2a  +  c. 
b  =  backlash  =  s  —  t. 
r= radius  of  fillet  at  root  of  tooth. 

For  gears  having  cast  teeth  the  following  are  usual  values: 

;    c=0.05p;    a=0.33p;    d=0. 


132  KINEMATICS  OF  MACHINERY. 

The  rims  of  such  gears  should  be  made  about  as  thick  as  the 
base  of  the  tooth  not  including  the  fillet.  The  hubs  are  usually 
about  twice  the  diameter  of  the  shaft.  When  the  arms  are  of 
elliptical  cross-section,  the  width  of  arm  at  the  inside  of  the  rim 
should  be  about  2J  times  the  circular  pitch.  Such  arms  are 
tapered  from  J"  to  f "  per  foot  on  each  side,  and  the  thickness  is 
made  equal  to  J  the  width. 

The  standard  proportions  of  cut  gear  teeth  are  expressed 
in  terms  of  the  diametral  pitch  as  follows: 

a=  !"-?-//;    6  =  0;     c  =  0.157" -s-p';    h=2.W7"+p'. 

The  radius  of  the  fillet  at  the  roots  of  cut  teeth  is  J  the 
width  of  the  space  between  the  teeth  measured  on  the  addendum 
circle.  The  outside  diameter  =  D  +  2a  =  (n  +  2)  -:-/>'. 

The  above  proportions  are  used  for  cut  gears  of  the  epicy- 
cloidal  and  14^°  involute  systems.  Where  greater  strength  is 
desired  without  a  corresponding  increase  in  the  pitch,  involute 
teeth  havjng  an  angle  of  obliquity  of  20°  are  sometimes  used. 
These  teeth  are  called  "  Stub  Teeth  "  because  they  are  made 
shorter  than  teeth  of  standard  proportions.  In  addition 
to  being  much  stronger,  these  teeth  have  less  sliding  and 
no  interference.  On  account  of  the  greater  obliquity  of  action 
the  normal  pressure  between  the  teeth  and  the  thrust  on 
the  bearings  are  greater  for  a  given  load  than  with  standard 
teeth. 

The  pitch  of  stub  teeth  is  usually  expressed  as  a  com- 
bination of  two  standard  diametral  pitches.  Thus,  a  4/5 
pitch  tooth  has  a  thickness  on  the  pitch  line  equal  to  that 
of  a  standard  4  pitch  tooth,  while  the  addendum  is  equal 
to  that  of  a  standard  5  pitch  tooth.  Other  pitches  are 
5/7,  6/8,  7/9,  8/10,  9/11,  10/12,  and  12/14.  The  depth  of 
the  space  inside  the  pitch  line  is  made  1J  times  the  ad- 
dendum. 


OUTLINES  OF  GEAR-TEETH. 


133 


74.  TJnsymmetrical  Teeth. — If  gears  are  always  to  turn  in  one 
direction,  the  opposite  sides  of  the  teeth  may  have  different  out- 
lines. Fig.  118  shows  such  teeth. 
The  working  sides  may  belong  to 
any  system  ;  the  backs  being  so 
formed  that  they  will  not  interfere, 
simply.  Involutes,  with  an  angle 
of  the  normal  greater  than  would 
be  practicable  for  working  faces  of 
teeth,  are  suitable  for  the  backs. 
Stronger  teeth,  for  any  pitch,  are 
obtained  by  this  construction. 

Gear-teeth  are  seldom  made  un-      \^  F'9- ll8 

symmetrical.  Such  forms  will  generally  be  more  expensive,  and 
the  necessary  strength  may  be  obtained  by  increasing  the  pitch.  If 
the  force  transmitted  is  excessive,  and  driving  is  always  in  one 
direction,  teeth  of  this  form  may  be  used  to  avoid  excessive  pitch. 

75.  Stepped  and  Twisted  Gearing. 
— The  action  of  gear-teeth  is  smooth- 
est when  the  contact  point  is  at  the 
line  of  centres;  for  in  this  phase 
there  is  pure  rolling  between  the 
teeth,  and,  in  the  epicycloid  system, 
the  obliquity  is  also  zero  at  this  in- 
stant. It  is  desirable  to  have  the  teeth 
as  short  as  the  required  arc  of  action 
permits;  but,  as  has  been  shown,  this 
is  governed  by  the  pitch,  which  is  a 
function  of  the  force  transmitted.  1 1 
is  therefore  unsafe  to  reduce  the  pitch 
beyond  a  certain  limit  in  a  given  case ; 
but  it  is  possible  by  "  stepped  "  teeth 
to  retain  the  required  pitch,  and 
still  have  a  pair  of  teeth  always 
in  contact  near  the  line  of  centres.  Suppose  a  spur  gear 


134  KINEMATICS  OF  MACHINERY. 

to  be  cut  by  a  series  of  equidistant  planes,  perpendicular  to  the 
axes ;  then  let  the  slices  into  which  the  gear  is  divided  be  placed 

as  in  Fig.  119.     If  there   are  N  of  these  slices,  each  maybe^th 

the  pitch  ahead  (or  behind)  the  adjacent  one.  In  this  arrangement, 
the  maximum  distance  of  the  nearest  contact  point  from  the  line 
of  centres  is  the  corresponding  distance  with  ordinary  spur-gears 
of  the  same  pitch  and  length  of  teeth,  divided  by  N. 

The  thickness  of  the  teeth  has  not  been  reduced  by  this  modi- 
fication, hence  the  strength  has  not  been  sacrificed.  Large  gears 
are  sometimes  constructed  on  this  principle,  with  two  sets  of  teeth, 
stepped  one-half  the  pitch. 

As  the  number  of  slices  into  which  the  gear  of  Fig.  119  is  cut 
increases  (their  thickness  decreasing  correspondingly)  the  teeth 
approach  those  of  Fig.  120,  which  represents  the  limiting  form  of 
the  stepped  wheels.  That  is,  when  the  number  of  slices  becomes 
infinite,  the  stepped  elements  become  spirals.  Gears  with  teeth  of 
this  kind  are  called  twisted  gears.  It  is  to  be  noticed  that  the 
action  of  these  twisted  teeth  is  similar  to  that  of  the  corresponding 
spur-gears,  and  they  must  not  be  confused  with  screw -gears  which 
they  resemble  in  form,  but  which  are  not  constructed  for  parallel 
axes.  The  distinction  will  be  considered  more  fully  in  a  later 
article.  With  twisted  gears  there  is  a  component  of  the  pressure 
transmitted  which  tends  to  slide  the  wheel  along  the  axis,  or  to 
crowd  the  shaft  to  which  the  wheel  is  attached  against  the  bearing. 
This  thrust  against  the  bearing  can  be  taken  up  by  a  collar,  and 
axial  motion  thus  prevented,  but  such  an  expedient  results  in  an 
undesirable  frictional  loss,  with  risk  of  heating,  etc.  By  twisting 

the  teeth  on  the  opposite  sides  of  the 
central  section  in  opposite  directions, 
as  shown  in  Fig.  120  (a),  the  axial 
efforts  due  to  these  two  halves 
Fig.  I20(a)  IB  balance  each  other,  and  there  is  no 

such  thrust  imparted  to  the  shaft.  In  actual  gears  of  this  form, 
the  two  halves  may  be  cast  in  one  piece,  if  the  teeth  are  not  to  be 


OUTLINES  OF  GEAR-TEETH.  135 

machined ;  but  if  cut  gears  are  used,  the  two  halves  are  made  as 
two  separate  gears  of  opposite  inclinations  (the  elements  of  one 
half  are  right-handed  helices,  and  of  the  other  are  left-handed 
helices),  and  these  two  gears  may  be  attached  firmly  to  the  shaft, 
side  by  side,  thus  constituting  practically  one  wheel. 

It  will  be  seen  that  these  twisted  gears  always  have  one  contact 
point  on  the  line  of  centres,  if  the  twist  within  the  width  (or  the 
half  width,  with  the  double  form  of  Fig.  120  (a))  of  the  gear  is  at 
least  equal  to  the  pitch,  and  the  action  at  this  one  point  is  pure 
rolling.  Now  if  the  addenda  of 
these  teeth  are  relieved  as  indi- 
cated in  Fig.  121  by  the  dotted 
lines,  the  faces  will  not  act  upon 
the  flanks  of  the  mating  gear  till 
the  pitch  point  of  any  section 
comes  into  contact;  that  is,  till 
the  line  of  centres  is  reached,  when  the  action  is  pure  rolling.  If 
the  mating  gear  is  similarly  treated,  the  two  gears  only  touch  at  a 
point,  and  this  point  is  always  in  the  line  of  centres.  Thus  contact 
for  any  tooth  begins  when  the  foremost  section  reaches  the  line  of 
centres  ;  it  travels  along  the  pitch  element  of  the  tooth  ending  as 
the  last  section  passes  the  line  of  centres,  the  most  advanced  sec- 
tion of  the  next  pair  of  teeth  taking  up  the  action  in  turn.  This 
concentrates  the  force  transmitted  at  a  single  point,  theoretically, 
which  may  result  in  too  intense  a  pressure  in  heavy  work  ;  but  it 
has  the  effect  of  producing  pure  rolling  between  the  teeth  in  con- 
tact at  all  times.  This  is  probably  the  only  example  of  combined 
pure  rolling,  constant  angular  velocity  ratio,  and  positive  driving. 

76.  Non-circular  Gears.— In  Arts.  46,  47,  48,  and  49  the  ac- 
tion of  rolling  ellipses,  rolling  logarithmic  spirals,  general  rolling 
curves,  and  lobed  wheels  was  briefly  explained.  It  has  been  seen 
that  cylinders  (in  the  general  sense)  corresponding  to  these  curves 
may  roll  together,  and  it  has  been  stated  that  such  surfaces  may  be 
used  as  pitch  surfaces  for  non-circular  gears. 

The  general  method  of  describing  gear-teeth,  as  given  in  Art 


13o  KINEMATICS  OF  MACHINERY. 

59,  may  be  applied  in  designing  teeth  for  such  gears,  but  a  con- 
venient approximate  method  will  be  indicated,  using  the  rolling 
ellipses  for  illustration.  Circular  arcs  can  be  drawn  which  closely 
approximate  the  ellipse  at  any  point,  and  the  methods  for  circular 
pitch  line  gears  can  then  be  used  for  the  teeth.  Or  accurate 
elliptical  curves  may  be  drawn  ;  then  lay  off  the  pitch  upon  them 
and  apply  the  method  given  in  Arts.  59  or  62,  in  generating 
the  teeth  on  these  arcs.  Of  course,  with  a  given  describing  curve, 
the  teeth  on  portions  of  the  pitch  line  which  have  different  curva- 
ture will  not  have  the  same  form. 

This  approximate  method  can  be  applied  to  other  rolling  curves 
as  well  as  to  the  ellipse,  and  it  is  thus  possible  to  form  teeth  for 
any  of  the  non-circular  forms  (including  the  lobed  wheels  of  article 
49)  which  will  transmit  motion  similar  to  that  due  to  the  rolling  of 
the  pitch  curves.  The  general  method  of  Art.  60  may  be  used  if 
preferred. 

77.  Approximate  Methods  of  Constructing  Profiles. — The  exact 
construction  of  the  tooth  profiles  is  somewhat  tedious,  and  in  many 
practical  applications  simpler  approximate  outlines  may  be  substi- 
tuted. Gears  with  cast  teeth,  especially  if  the  pitch  is  small,  de- 
part somewhat  from  the  ideal  form,  however  carefully  the  patterns 
may  be  made  ;  and,  therefore,  some  one  of  the  approximate  methods 
is  generally  used  for  laying  out  the  patterns.  In  making  cutters  for 
cut  gears,  the  exact  method  is  usually  employed.* 

The  arcs  of  the  curves  (epicycloids  and  hypocycloids,  or  invo- 
lutes) used  in  gear-teeth  are  so  short  that  circular  arcs  can  be 
found  which  very  closely  approximate  these  curves  ;  and  most  of 
the  approximate  constructions  are  circular-arc  methods. 

A  method  given  in  Unwinds  Machine  Design  has  the  merit  of 
requiring  no  tables  or  special  instruments,  and  it  will  be  described 
first.  In  Fig.  122,  A  and  B  are  the  two  pitch  circles,  and  G  and 

*  A  little  book  published  by  the  Pratt  &  Whitney  Co.  gives  a  description  of 
the  machine  used  by  this  company  for  accurately  making  these  cutters  auto- 
matically. This  treatise  was  written  by  Professor  McCord,  and  is  reproduced 
in  his  Kinematics. 


OUTLINES  OF  GEAR-TEETH.  137 

G'  are  the  describing  circles.  Ph  is  the  height  of  the  addendum 
of  Bt  and  Pe  represents  two  thirds  this  height.  If  a  circle  is 
drawn  through  e  with  a  centre  at  the  centre  of  the  pitch  circle  B, 
it  cuts  the  describing  circle  G  in  g\  and  if  the  arc  Pb,  on  the  pitch 
circle  B,  is  laid  off  equal  to  the  arc  Pg  on  the  generating  circle  G, 


b  and  g  are  two  points  in  the  tooth  outline,  as  in  the  exact  con- 
struction. Draw  gP,  the  normal  to  the  tooth  profile  at  g.  Now 
find  by  trial  a  circular  arc  with  a  centre  on  Pg,  or  its  extension 
beyond  P,  which  will  pass  through  g  and  b.  This  arc  passes  through 
two  points  of  the  exact  tooth  outline,  and  its  tangent  at  g  also  corre- 
sponds in  direction  with  that  of  the  true  epicycloid,  as  the  normal 
at  this  point  is  Pg  for  both  the  exact  and  the  approximate  curves. 
If  the  arc  Pa  is  laid  off  on  the  pitch  circle  A,  equal  to  the  arc  Pg, 
another  circular  arc,  with  a  centre  on  the  normal  Pg,  will  pass 
through  a  and  g,  and  it  will  approximate  the  flank  of  A.  By  a 
similar  method,  Pe'  is  laid  off  equal  to  two  thirds  the  height  of  the 
addendum  of  A,  and  a  circular  arc  with  a  centre  on  Pg',  and  passing 
through  a'  and  g',  is  an  approximation  to  the  exact  face  profile  of 
A.  The  approximate  outline  for  the  flank  of  B  is  an  arc  passing 
through  b'  and  g',  with  a  centre  on  g'P,  or  its  extension.  The  point 
e  need  not  necessarily  be  two  thirds  of  Ph ;  but  this  fraction  gives 
a  good  distribution  of  the  error. 

A  method  due  to  Professor  Willis  has  been  more  widely  used, 
perhaps,  than  any  other.     It  may  be  briefly  explained  as  follows: 


138  KINEMATICS  OF  MACHINERY. 

Lay  off  from  P  (Fig.  123)  the  arc  Pa  =  Pa'  —  one-half  the  pitch, 
and  draw  radii  Oa  and  Oa'\  then  draw  the  lines  mm  and  m'm' 
through  a  and  a',  making  an  angle  0  with  Oa  and  Oa'  respect- 
ively. At  a  point  c  (on  mm)  take  a  centre,  and  draw  a  circular  arc 
Pf  through  P;  also  with  a  centre  at  c'  (on  m'm')  draw  the  arc  Pf 
through  P.  The  angle  0  and  the  centres  c  and  c'  may  be  so  chosen 
that  these  arcs  will  have  radii  of  curvature  equal  to  the  mean  radii 
of  curvature  of  the  proper  epicycloidal  and  hypocycloidal  faces  and 


flanks  of  the  tooth.  The  angle  0  was  found  by  the  originator  of 
this  method  to  give  the  best  results  when  taken  at  75°;  a .  .  c  and 
a' . .  c'  are  given  by  the  following  formulas,  in  which p  =  pitch,  and 

n  ==  number  of  teeth  in  the  wheel:  a  .  .  c  f  or  faces  =  "ff      .    -10)  > 

a' .  .  c'   for  flanks  =  -^-f — -j.       An     instrument    known    as 

2  \n  —  I'ZJ 

Willis's  Odontograph  facilitates  these  operations.  The  form  of 
this  instrument  is  indicated  by  the  dotted  lines  of  Fig.  123.  It  is 
graduated  along  the  edge  m'm'  each  way  from  the  point  «',  and  a 
table  which  accompanies  the  instrument  gives  the  positions  of  the 
centres  c  and  c'  in  terms  of  these  graduations  for  wheels  of  given 
numbers  of  teeth  and  pitch. 


OUTLINES  OF  GEAR-TEETH. 


139 


Mr.  George  B.  Grant  has  improved  upon  this  odontograph  by 
tabulating  the  radii  for  faces  and  flanks,  and  also  tabulating  the 
radial  distances  of  the  centres  c  and  c'  from  the  pitch  circle.  Mr. 
Grant  has  compiled  a  similar  set  of  tables,  called  the  Grant  Odonto- 
graph, which  are  used  in  precisely  the  same  way;  but  the  values  are 
different,  giving  somewhat  different  tooth  profiles  from  those  of  the 
Willis  system.  The  Willis  method  gives  circular  arc  tooth  outlines 
which  are  correct  for  one  point,  and  which  have  radii  equal  to  the 
mean  radius  of  curvature  of  the  exact  curve.  The  approximate 
faces  derived  by  this  system  lie  entirely  within  the  true  epicycloids. 
Mr.  Grant's  system  gives  arcs  which  pass  through  three  points  of 
the  exact  profiles  of  the  faces,  and  thus  more  closely  approximate 
the  correct  curves. 

The  application  of  Grant's  Cycloidal  Odontograph,  or,  as  its 
author  calls  it,  from  the  method  of  deriving  it,  the  Three-Point 
Odontograph,  is  shown  in  Fig.  124.  The  accompanying  table  is 


Fig.  124 


given,  together  with  his  table  for  constructing  approximate  involute 
teeth.  This  matter  is  taken  from  Odontics,  or  the  Teeth  of  Gears, 
by  George  B.  Grant,  and  is  reproduced  by  permission  of  the  author. 
To  use  the  table  in  drawing  approximate  epicycloidal  teeth  pro- 
ceed as  follows:  Draw  the  pitch  line  and  set  off  the  pitch,  dividing 
the  latter  properly  for  thickness  of  tooth  and  space.  The  table 
gives  values  both  in  terms  of  One  Diametral  Pitch  (equal  3.14" 


140 


KINEMATICS  OF  MACHINERY. 


circular  pitch),  and  of  One  Inch  Circular  Pitch.     Use  the  part  of 
the  table  corresponding  to  the  system  of  pitch  employed. 

THREE-POINT  ODONTOGRAPH. 

STANDARD   CYCLOIDAL   TEETH.      INTERCHANGEABLE   SERIES. 
(From  Geo.  B.  Grant's  "  Odontics.") 


For  One  Diametral  Pitch. 

For  One  Inch  Circular  Pitch. 

Number  of  Teeth. 

For  any  other  pitch  divide  by  that 
pitch. 

For  any  other  pitch  multiply  by 
that  pitch. 

Faces. 

Flanks. 

Faces. 

Flanks. 

Exact. 

Intervals. 

Bad. 

Dis. 

Rad. 

Dis. 

Rad. 

Dis. 

Rad. 

Dis. 

10 

10 

1.99 

.02 

-  8.00 

4.00 

.62 

-01 

-2.55 

1.27 

11 

11 

2.00 

.04 

-11.05 

6.50 

.63 

.01 

-3.34 

2.07 

12 

12 

2.01 

.06 

oo 

CO 

.64 

.02 

GO 

CO 

is* 

13-14 

2.04 

.07 

15.10 

9.43 

.65 

.02 

4.80 

3.00 

15| 

15-16 

2.10 

.09 

7.86 

3.46 

.67 

.03 

2.50 

1.10 

m 

17-18 

2.14 

.11 

6.13 

2.20 

.68 

.04 

1.95 

0.70 

20 

19-21 

2.20 

.13 

5.12 

1.57 

.70 

.04 

1.63 

0.50 

23 

22-24 

2.26 

.15 

4.50 

1.13 

.72 

.05 

1.43 

0.36 

27 

25-29 

2.33 

.16 

4.10 

0.96 

.74 

.05        1.30 

0.29 

33 

30-36 

2.40 

.19 

3.80 

0.72 

.76 

.06 

1.20 

0.23 

42 

37-48 

2.48 

.22 

3.52 

0.63 

.79 

.07 

1.12 

0.20 

58 

49-72 

2.60 

.25 

3.33 

0.54 

.83 

.08 

1.06 

0.17 

97 

73-144 

2.83 

.28 

3.14 

0.44 

.90 

.09 

1.00 

0.14 

290 

145-300 

2.92 

.31 

3.00 

0.38 

.93 

.10 

0.95 

0.12 

00 

Rack 

2.96 

.34 

2.96 

0.34 

.94 

.11 

0.94 

0.11 

The  example  of  Fig.  124  is  a  wheel  of  20  teeth,  2  diametral 
pitch,  hence  the  pitch  circle  is  10  inches  diameter  (this  figure  is 
not  reproduced  full  size).  Opposite  20  teeth  in  the  table  we  find 
.13  in  the  column  of  distances  for  faces  ("dis.") ;  divide  this  by  the 
diametral  pitch  (2),  giving  .06"  as  the  distance  of  the  circle  of  face 
centres  from  the  pitch  circle.  Lay  this  distance  off  inside  the 
pitch  circle,  and  draw  a  circle  through  this  point,  concentric  with 
the  pitch  circle.  In  a  similar  way  the  distance  for  flanks  (1.57)  is 
divided  by  2,  giving  .78",  which  is  laid  off  outside  the  pitch  circle, 
and  a  circle  is  drawn  through  this  point.  All  tooth  faces  are  to  be 


OUTLINES  OF  GEAR-TEETH.  141 

drawn  with  circular  arcs  having  centres  on  the  first  of  these  lines 
of  centres,  and  the  flanks  are  drawn  by  arcs  having  centres  on  the 
last  found  line  of  centres.  The  tabular  radius  of  faces  for  20  teeth 
is  given  as  2.20,  and  dividing  this  by  the  diametral  pitch,  we  get 
1.10"  as  the  radius  for  the  faces  of  the  2-pitch  wheel.  With  this 
radius  and  centres  on  the  face  centre  line,  draw  arcs  through  the 
proper  points  in 'the  pitch  circle,  of  course  having  the  concave 
sides  of  the  arcs  toward  the  body  of  the  teeth.  In  a  similar  way, 
the  tabular  radius  im  flanks  (5.12)  is  divided  by  the  diametral  pitch, 
giving  2.56"  as  the  corrected  radius.  With  centres  on  the  flank 
centre  line,  draw  arcs  with  this  radius  meeting  the  face  arcs  already 
drawn  at  the  pitch  point,  and  with  the  concave  sides  towards  the 
spaces.  Terminate  the  tooth  profiles  by  the  addendum  and  root 
circles,  determined  as  in  Art.  73,  and  put  in  fillets  at  the  bottoms 
of  the  spaces. 

If  the  circular  pitch  is  used  the  construction  is  similar,  using 
the  appropriate  portion  of  the  table,  but  multiplying  the  tabular 
values  by  the  circular  pitch  in  inches  instead  of  dividing. 

As  explained  above,  this  table  is  calculated  for  arcs  which  pass 
through  three  points  in  the  true  curve.  It  is  recommended  that 
the  student  construct  tooth  profiles  on  a  large  scale  by  the  exact 
method,  and  then  draw  the  approximate  profiles  (superimposed), 
for  comparison. 

Grant's  involute  Odontograph  given  on  page  142  is  used  as  fol- 
lows :  Lay  off  the  pitch  circle,  addendum,  root  and  clearance  lines, 
as  in  the  preceding  case.  "  Draw  the  base  line  one  sixtieth  of  the 
pitch  diameter  inside  the  pitch  line.  Take  the  tabular  face  radius 
on  the  dividers,  after  multiplying  or  dividing  it  as  required  by  the 
table,  and  draw  in  all  the  faces  from  the  pitch  line  to  the  addendum 
line  from  centres  on  the  base  line.  Set  the  dividers  to  the  tabular 
flank  radius  (corrected),  and  draw  in  all  the  flanks  from  the  pitch 
line  to  the  base  line.  Draw  straight  radial  flanks  from  the  base 
line  to  the  root  line,  and  round  them  into  the  clearance  line. '' 
[Grant's  Teeth  of  Gears,  p.  30.] 


142 


KINEMATICS  OF  MACHINERY. 


INVOLUTE  ODONTOGRAPH. 
STANDARD  INTERCHANGEABLE  TOOTH,  CENTRES  ON  THE  BASE  LINE. 


Teeth. 

Divide  by  the 
Diametral  Pitch. 

Multiply  by  the 
Circular  Pitch. 

Teeth. 

Divide  by  the 
Diametral   Pitch. 

Multiply  by  the 
Circular  Pitch. 

Face 
Radius. 

Flank 
Radius. 

Face 
Radius. 

Flank 
Radius. 

Face 
R  dins. 

Flank 
Radius. 

Face 
Radius. 

Flank 
Radius. 

10 

2.28 

.69 

.73 

.22 

i       28 

3.92 

2.59 

1.25 

0.82 

11 

2.40 

.83 

.76 

.27 

29 

3.99 

2.67 

1.27 

0  85 

12 

2.51 

.96 

.80 

.31 

30 

4.06 

2.76 

1.29 

0.88 

13 

2.62 

1.09 

.83 

.34 

31 

4.13 

2.85 

1.31 

0.91 

14 

2.72 

1.22 

.87 

.39 

32 

4.20 

2.93 

1.34 

0.93 

15 

2.82 

1.34 

.90 

.43 

33 

4.27 

3.01 

1.36 

0.96 

16 

2.92 

1.46 

.93 

.47 

34 

4.33 

3.09 

1.38 

0.99 

ir 

3.02 

1.58 

.96 

.50 

35 

4.39 

3.16 

1.39 

1.01 

18 

3.12 

1.69 

.99 

.54 

36 

4.45 

3.23 

1.41 

1.03 

19 

3.22 

1.79 

1.03 

.57 

37-40 

4.20 

1.34 

20 

3.33 

1.89 

1.06 

.60 

41-45 

4.63 

1.48 

•21 

3.41 

1.98 

1.09 

.63 

46-51 

5.06 

1.61 

22 

3.49 

2.06 

1.11 

.66 

52-60 

5.74 

1.83 

23 

3.57 

2.15 

1.13 

.69 

61-70 

6.52 

2.07 

24 

3.64 

2.24 

1.16 

.71 

71-90 

7.72 

2.46 

25 

3.71 

2.33 

1.18 

.74 

[    91-120 

9.78 

3.11 

26 

3.78 

2.42 

1.20 

.77 

121-180 

13.38 

4.26 

27 

3.85 

2.50 

1.23 

.80 

181-360 

21.62 

6.88 

Grant's  special  directions  for  drawing  the  teeth  of  the  involute 
rack  are  substantially  as  follows:  To  draw  the  teeth  for  the  involute 
rack,  draw  lines  at  75°  with  the  pitch  line  of  the  rack;  the  outer 
quarter  of  the  tooth  length  (one  half  the  addendum)  is  to  be 
rounded  off  by  an  arc  with  a  radius  equal  to  240"  divided  by  the 
diametral  pitch,  or  .67"  multiplied  by  the  circular  pitch.  This  is 
to  avoid  interference. 

78.  Bevel-gears.  -  It  was  shown  (see  Art.  52)  that  a  pair  of  cones 
can  "be  placed  on  intersecting  axes  in  such  a  manner  that  they  will 
transmit  motion  with  a  given  angular  velocity  ratio  if  they  roll  to- 
gether without  slipping.  Such  rolling  cones  may  be  used  for  pitch 
surfaces  of  bevel-gears  just  as  rolling  cylinders  are  used  for  the 
pitch  surfaces  of  spur-gears,  and  teeth  can  be  formed  on  these 
conical  pitch  surfaces  which  will  transmit  a  positive  motion  equiva- 
lent to  that  of  the  rolling  cones. 


OUTLINES  OF  GEAR-TEETH. 


143 


In  treating  of  spur-gearing  plane  sections  at  right  angles  to  the 
axes  were  used  to  represent  the  gear,  and  the  tooth  outlines  were 
considered  to  be  developed  by  the  rolling  of  one  plane  curve 
(the  describing  line)  upon  another  plane  curve  (the  pitch  line). 
The  real  teeth  are,  of  course,  solids  bounded  by  ruled  surfaces, 
all  transverse  sections  of  which  are  exact  counterparts  of  the 
plane  curves  discussed.  That  is,  the  actual  teeth  are  not  lines 
generated  by  a  point  in  the  describing  curve  as  it  rolls  upon  the 
pitch  line,  but  they  are  really  surfaces  generated  by  an  element  of 
the  describing  cylinder  as  it  rolls  upon  the  pitch  cylinder. 

Spur-gears  coming  under  the  head  of  plane  motions  permit 
representation  by  plane  sections,  as  explained  in  Art.  10.  This 
simple  treatment  cannot  be  applied  to  bevel-gears,  for  although 
each  separate  gear  has  a  plane  motion  (rotation)  about  its  axis, 
taking  two  cones  rolling  together,  the  relative  motion  is  a  spherical 
motion  (see  Art.  12).  To  construct  bevel  gear-teeth  two  projections 
are  required. 

Just  as  an  element  of  one  cylinder  in  rolling  upon  another  cylin- 


Fig.  125. 


tier  generates  a  tooth  surface,  so  an  element  of  one  cone  in  rolling 
upon  another  cone  sweeps  up  a  surface  which  can  be  used  as  the 
basis  of  a  bevel-gear  tooth.  Fig.  125  shows  a  pitch  cone  A  with 
the  generating  cone  G  (of  equal  slant  height)  in  contact  with  it 


144 


KINEMATICS  OF  MACHINERY. 


along  a  common  element  PQ.  All  points  in  the  bases  of  both  cones 
are  at  the  same  distance  from  the  common  apex,  hence  these  bases 
are  small  circles  of  a  sphere  which  has  a  radius  equal  to  the  common 
slant  height.  If  the  generating  cone  be  rolled  upon  the  pitch  cone, 
a  point  in  the  base  of  Gr,  as  g,  will  describe  a  curve  on  the  surface  of 
the  sphere,  relative  to  the  base  of  the  pitch  cone.  This  curve 
is  analogous  to  the  epicycloid,  the  derivation  of  which  was  treated  in 
Art.  62,  and  the  curve  now  under  consideration  may  be  called  a 
spherical  epicycloid.  In  a  similar  manner  another  generating  cone 
G'  can  roll  inside  the  pitch  cone,  a  point  g'  in  its  base  tracing  a 
spherical  hypocycloid  on  the  surface  of  the  sphere.  Points  on  other 
transverse  sections  of  these  generating  cones  would  trace  similar 
curves  on  spheres  of  different  radii.  A  line  passing  through  the 
centre  of  the  sphere  (the  common  apex  of  the  cones)  and  moving 
along  the  spherical  epicycloids  and  hypocycloids  described  as  above, 
would  give  surfaces  portions  of  which  could  be  used  as  tooth 
boundaries.  In  other  words,  the  elements  of  the  describing  cones 

which  pass  through  g  and  g'  would 
sweep  up  these  surfaces.  If  two  pitch 
cones  of  equal  slant  height  have  teeth 
generated  in  the  manner  just  outlined, 
they  will  work  together  properly,  trans- 
mitting a  positive  motion  equivalent  to 
the  rolling  of  the  pitch  surfaces,  pro- 
vided the  pitch  of  the  teeth  agree,  and 
that  the  faces  of  each  wheel  are  de- 
scribed by  the  same  generating  cone 
/]  which  describes  the  flanks  of  the  other 
wheel. 

Fig.  126  indicates  two  pitch  cones  A 
and  B,  with  axes  QO  and  QO',  respect- 
ively, and  a  common  element  of  contact 
PQ.     The  describing  cones  are  G  and 
Fig'126-  £',  with  axes   Qo  and   Qo';  and  if   all. 

four  axes  lie  in  one  plane  (a  meridian  plane  of  the  sphere),  they 


OUTLINES  OF  GEAR-TEETH.  145 

can  roll  together  about  fix.ec!  axes,  always  having  a  common  con- 
tact element  in  PQ.  As  the  rolling  proceeds,  such  an  element  of  G 
asgQ  sweeps  up  the  faces  for  B  and  the  flanks  for  A;  and,  at  the 
same  time,  the  element  g' Q  of  G'  sweeps  up  the  faces  of  A  and 
the  flanks  of  B. 

The  faces  of  B  and  the  flanks  of  A  always  have  gQ  for  a 
common  element,,  and  a  plane  through  the  three  points  PgQ> 
the  common  normal  plane  at  the  element  of  contact  of  these 
surfaces,  always  passes  through  the  contact  element  of  the 
pitch  cones  PQ.  Likewise,  the  normal  plane  to  the  faces  of  A 
and  the  flanks  of  B,  Pg'Q,  always  passes  through  PQ',  hence  teeth 
bounded  by  these  swept-up  surfaces  will  transmit  motion  with  a 
constant  angular  velocity  ratio.  The  analogy  between  this  case  and 
that  of  the  spur-gear  teeth,  treated  in  Art.  62  (Fig.  110),  is  so  close 
that  further  discussion  is  hardly  necessary. 

The  describing  surfaces  are  not  necessarily  right  cones  of  circu- 
lar cross-section,  though  these  are  the  figures  which  correspond  to 
the  epicycloidal  class  of  spur-gears,  and  are  the  only  forms  com- 
monly employed.  Any  cone  with  an  apex  at  the  common  apex  of 
the  pitch  cones,  and  tangent  to  them  along  their  common  element 
might  be  used,  as  it  would  satisfy  the  kinematic  requirements. 

Involute  teeth  for  bevel  gears  maybe  generated  in  a  manner  anal- 
ogous to  that  used  in  Art.  (58  for  the  generation  of  spur  gear  teeth. 

It  is  difficult  to  construct  spherical  epicycloids  and  hypocycloids, 
and  to  represent  the  teeth  of  bevel  gears  on  paper,  arid  in  practice 
a  method  know  as  Tredgold's  approximation  is  always  employed. 

79.  Tredgold's  Approximate  Method  of  Drawing  Bevel-gear 
Teeth. — Fig.  127  shows  the  projection  of  two  cones  (with  bases  PM 
and  PN)  on  a  plane  parallel  to  both  axes  QO  and  QO'.  The  line  00' 
is  drawn  perpendicular  to  the  contact  element  PQ,  then  OM  is  drawn 
perpendicular  to  MQ  and  O'N  is  drawn  perpendicular  to  NQ.  A  cone 
can  be  constructed  on  the  axis  OQ  with  OP  and  OM  as  elements, 
and  another  on  O'Q  with  O'P  and  O'N  as  elements.  These  cones 
are  called  normal  cones  to  A  and  J5,  respectively,  as  any  element  of 
one  of  these  cones  is  perpendicular  to  an  element  of  the  pitch  cone 


146 


KINEMATICS  OF  MACHINERY. 


having  the  same  axis  and  the  same  -base.  The  surfaces  of  these 
noimal  cones  approximate  the  spherical  surface  for  a  short  space 
either  side  of  the  pitch  circles,  and  the  conical  surfaces  have  the 
practical  advantage  that  they  can  be  developed  upon  a  plane  for 
the  construction  of  tooth  profiles.  Tredgold's  approximate  method 
consists  in  describing  tooth  outlines  on  these  developed  surfaces  of 
the  normal  cones,  and  then  wrapping  these  surfaces  back  to  their 
original  positions.  The  development  of  the  normal  cone  surfaces 
is  indicated  in  Fig.  127  by  PM'O,  and  PN'O'.  Upon  the  de- 


Fig.  127. 


veloped  bases  of  these  cones  (PM'  and  PNf)  as  pitch  lines,  tooth 
outlines  can  be  drawn  by  any  of  the  methods  used  for  spur-gears, 
just  as  if  these  were  the  pitch  lines  of  such  gears;  and  when  these 
surfaces  are  rolled  back  into  the  normal  cones  the  ends  of  the  teeth 
are  given  by  the  profiles  constructed  in  this  way.  A  straight  line 
passing  through  Q  and  following  such  a  profile  would  sweep  up 
tooth  surfaces,  all  elements  of  which  are  right  lines  converging  at 
Q.  Within  the  limits  of  practice,  such  teeth,  if  properly  con- 
structed, agree  quite  closely  with  the  exact  forms. 

The  application  of  this  method  is  shown  in  detail  in  Fig.  128 
for  the  teeth  of  a  single  wheel.  The  pitch  cone  is  shown  in  side 
elevation  by  MQM,  and  in  plan  by  the  circle  M^Mi.  The  side  ele- 
vation of  the  normal  cone  is  projected  in  MOM,  and  the  develop- 


OUTLINES  OF  GEAR-TEETH. 


147 


ment  of  the  pitch  circle  is  show  by  MM'.  The  pitch,  which  must 
be  an  aliquot  part  of  the  pitch  circle  MlMl  (  =  MM  times  n),  is 
laid  off  on  MM',  and  the  addendum  and  root  circles  (AAf  and 
RR',  respectively)  are  drawn  to  give  the  proper  length  of  teeth. 
The  tooth  outline  is  now  constructed  on  MM'  as  for  a  spur-gear. 

It  is  evident  that  all  of  the  pitch  points  will  fall  upon  the  line 
MM,  in  the  side  elevation,  when  the  normal  cone  surface  is  re- 


Fig.  128. 


turned  to  its  original  position;  that  the  outer  ends  of  the  teeth  will 
fall  upon  A  A;  and  that  the  bottoms  will  fall  upon  RR.  In  the 
other  projection,  the  pitch  points  will  all  lie  on  the  circle  MiMi; 
the  tops  will  fall  on  the  circle  through  A^A^  and  the  bottoms  will 
be  in  the  circle  RiR^  In  this  last  projection  (plan)  the  teeth  will 
all  appear  the  same,  and  they  will  have  their  true  thickness  at  all 
parts;  but  the  height  (AR]  will  be  shortened  to  AiR,.  Divide  up 
the  circle  J/, Mi  into  the  proper  number  of  divisions  for  teeth  and 


148  KINEMATICS  OF  MACHINERY. 

spaces,  and  draw  radii  through  the  middle  of  the  tooth  divisions. 
Lay  off  half  the  thickness  of  the  tooth  at  the  pitch  line  (as  obtained 
from  the  construction  on  developed  pitch  line  MM')  each  side  of 
these  middle  radii  upon  the  circle  M^Mj  then  lay  off  half  the 
thickness  of  the  top  in  a  similar  way  on  the  circle  A^A^  and  half 
the  thickness  at  the  bottoms  on  the  circle  R1R1.  The  half  thick- 
ness at  positions  intermediate  between  the  pitch  circle  and  the  ad- 
dendum or  root  circles  can  also  be  laid  off  on  the  corresponding 
circles  in  the  plan,  taking  as  many  such  thicknesses  as  the  desired 
accuracy  requires.  The  method  of  finding  these  intermediate 
points  is  indicated  in  Fig.  128.  Through  the  points  thus  found,  the 
curves  A1M1R1  can  be  drawn,  giving  the  plan  of  the  large  ends  of 
the  teeth.  To  complete  the  plan  of  the  wheel  we  may  proceed  as 
follows:  Draw  radii  from  Av  Ml9  R^,  etc.,  to  Q^  then  lay  off  Aa, 
on  the  side  elevation,  equal  to  the  desired  length  of  the  tooth,  or 
the  face  of  the  gear,  and  draw  amr  parallel  to  AMR;  from  a,  m,and 
r,  points  lying  in  elements  from  Q  to  A,  M9  and  R,  respectively, 
carry  lines  across  parallel  to  QQt;  then  circles  with  Qlf  as  a  centre 
and  tangent  to  these  several  parallels  are  the  projections  in  the 
plan  of  the  addendum,  pitch,  and  root  circles  for  the  small  end  of 
the  teeth.  As  all  elements  of  the  teeth  converge  in  plan  at  Qlf  the 
intersections  of  radii  through  Al9  Ml9  and  7£,  with  these  circles  last 
drawn  locate  points  al9  ml9  and  rl  in  the  plan  of  the  small  ends  of 
the  teeth.  Curves  through  these  intersections,  with  the  portions  of 
the  radial  elements  intercepted  between  them  and  the  outer 
curves,  will  complete  this  projection  of  the  teeth.  Returning  to 
the  side  elevation,  the  lines  aa,  mm,  and  rr,  are  the  projections  of 
the  smaller  addendum,  pitch,  and  root  circles.  To  complete  the 
side  elevation  of  the  teeth  project  across  (parallel  to  Q^)  from  the 
various  points  on  A^A^  to  AA\  from  M}M^  to  MM;  from  R^R^  to 
RR,  etc.,  and  draw  curves  through  the  intersections  thus  made. 
This  will  give  the  side  elevation  of  the  large  ends  of  the  teeth  by 
passing  curves  through  the  corresponding  intersections.  In  a 
similar  way  the  side  elevation  of  the  smaller  ends  is  obtained,  and 
elements  through  Aa,  Mm,  Rr,  etc.,  completes  this  view.  It  is 


OUTLINES  OF  GEAR-TEETH.  149 

evident  that  each  of  the  teeth  appears  in  this  projection  with  its 
own  distinctive  form. 

The  ring,  or  rim,  which  supports  the  teeth  usually  has  a  thick- 
ness equal  to  the  roots  of  the  teeth  at  the  large  ends,  and  this  rim, 
with  the  hub,  arms,  etc.,  can  now  be  drawn. 

80.  Peculiar  Smoothness  in  Operation  of  Bevel-gearing.— Among 
spur-gears  of  an  interchangeable  system,  those  with  the  larger  pitch 
circles   will    drive   more    smoothly,   other   conditions    being    the 
same.     By  referring  to  the  bevel-gear  of  Fig.  128  it  will  be  seen 
that  there  are  16  teeth  on  a  pitch  circle  of  radius  QiMi  (diameter 
=  MM}}  but  these  teeth  have  profiles  similar  to  those  of  a  spur- 
gear   with  a  pitch  radius   OM,  equal  to  the  slant  height  of  the 
normal  cone,  and  therefore  the  action  of  the  bevel-gear  would  cor- 
respond to  that  of  this  larger  spur-gear  instead  of  to  a  spur-gear  of 
diameter  MM. 

The  actual  pitch  diameter  of  the  bevel-gear  is  to  that  of  the 
equivalent  spur-^ear  (so  far  as  smoothness  of  running  is  con- 
cerned) as  sin  AOQ :  1 ;  thus  if  the  pair  of  bevel-gears  on  shafts  at 
right  angles  are  equal,  angle  AOQ  =  45°,  when 

sin  AOQ :  1 : :  i\/2 : 1 : :  .707 : 1. 

81.  Non- inter  changeability  of  Bevel-gears.  —  Bevel-gears    are 
almost  always  made  to  work  together  in  pairs,  and  it  is  not  there- 
fore of  great  importance  to  adopt  a  standard  describing  circle  for 
all  pairs  of  the  same  pitch.     If  two  intersecting  axes  approach 
each  other  at  a  fixed  angle,  there  is  but  one  bevel-gear  which  will 
work  properly  with  any  other  gear;*  for  a  change  in  the  angular 
velocity  ratio  involves  a  change  in  the  direction  of  the  contact  ele- 
ment (Ac  of  Fig.  96),  and  hence  a  change  in  both  pitch  cones.     A 
given  bevel-gear  could  work  with  more  than  one  other  wheel  if  the 
inclination  of  the  axes  varied  correspondingly,  but  this  is  a  con- 

*  Mr.  Hugo  Bilgram  has  produced  sets  of  bevel-gears,  by  his  gear-shaper, 
in  which  several  different  sizes  of  gears  tvork  correctly  with  a  single  gear,  and 
all  the  axes  make  the  same  angle  with  the  common  driver.  However,  the 
pitch  cones  all  of  these  gears  do  not  have  a  common  apex,  although  the  teeth 
elements  all  converge  to  a  common,  point.  These  gears  are 'not,  properly;  of 
the  common  type  of  bevel-gears. 


150  KINEMATICS  OF  MACHINERY. 

dition  seldom  met,  and  so  these  gears  may  generally  be  designed 
to  work  in  pairs  without  regard  to  other  gears. 

The  describing  circles  (if  the  epicycloidal  system  is  used)  may 
be  so  taken  that  both  wheels  will  have  radial  flanks,  which  gives 
a  simple  form  of  teeth  to  construct,  though  this  is  not  always  de- 
sirable. For  convenience  of  manufacture  it  is  desirable  to  have  a 
uniform  system,  usually;  and  when  bevel-gears  are  cut  with  rotary 
milling  cutters  of  the  common  type,  standard  cutters  are  used  for 
various  gears  of  the  same  pitch.  This  will  be  discussed  in  a  later 
article. 

In  a  great  majority  of  cases  requiring  bevel-gears  the  axes  are 
at  right  angles  to  each  other,  and  "  stock-gears  "  for  such  cases  can 
frequently  be  obtained  from  gear-makers,  if  the  proportions  are 
not  unusual.  These  stock-gears  are  generally  much  less  expensive 
than  gears  made  to  order;  but  special  gears  are  almost  always 
required  when  the  angle  of  the  axes  is  other  than  90°.  When  the 
two  gears  are  equal  (angular  velocity  ratio  1:1),  the  gears  are- 
called  Mitre  Gears. 

82.  Helical  Gears. — If  two  cylinders  are  tangent  to  each  other,, 
they  will  have  line  contact  when  the  axes  are  parallel,  and  point 
contact  when  the  axes  are  not  parallel.  In  the  latter  case  the 
two  elements  (one  of  each  cylinder)  through  the  point  of  contact 
determine  a  plane  tangent  to  both  cylinders.  The  radii  of  both, 
cylinders  at  the  point  of  contact  are  perpendicular  to  this  plane. 
When  the  cylinders  rotate  on  their  axes,  each  of  the  points  in  con- 
tact moves  in  the  tangent  plane  in  a  direction  at  right  angles  to 
the  axis  of  rotation.  This  is  shown  in  Fig.  129,  which  represents 
a  plan  view  of  two  cylinders,  A  and  5,  in  contact  at  P ;  a  ...  a 
and  b  ...  b  being  the  axes  of  the  respective  cylinders.  The  angle 
between  the  axes  of  the  cylinders  is  6.  The  linear  velocities  of 
the  points  of  A  and  B  in  contact  at  P  are  assumed  to  be  v,  and 
v-t  respectively.  vt  is  represented  by  PZ,  and  v2  by  Pm.  These 
velocities  have  a  common  component,  v,  represented  by  Pn,  drawn 
through  P  perpendicular  to  Im.  The  angle  between  v  and  v^  is 
a,  that  between  v  and  i\  is  /?.  The  components  of  vt  and  v2,  in 


OUTLINES   OF  GEAR-TEETH. 


151 


a  direction  perpendicular  to  v  are  Pt  and  Pi'  respectively. 
The  algebraic  difference  of  these  components  represents  sliding 
at  the  points  of  contact,  while 
the  common  component  rep- 
resents rolling  contact  in  a 
direction  perpendicular  to  the 
sliding.  It  appears  that  the 
action  of  the  tangent  cylinders 
may  be  considered  as  a  com- 
bination of  sliding  in  a  direc- 
tion, it',  determined  by  the  re- 
lative magnitudes  of  the  linear 
velocities  of  the  contact  points, 
and  rolling  in  a  direction,  Pn,  Fifl>  I29 

perpendicular  to  the  sliding. 

Considering  the  rolling  and  the  sliding  separately,  it  is  evident 
that,  with  a  constant  angular  velocity  ratio  between  the  cylinders 
corresponding  to  the  assumed  linear  velocities,  the  rolling  action 
will  result  in  rolling  between  two  helices,  each  of  which  is  called 
a  normal  helix,  crossing  the  elements  of  the  respective  cylinders 
at  angles  of  90°  —  a  and  90°— /?,  while  the  sliding  is  between  helices 
crossing  the  elements  at  angles  of  a  and  /?  respectively. 

If  these  cylinders  are  used  as  the  pitch  surfaces  of  gears  to 
have  an  angular  velocity  ratio  corresponding  to  the  assumed 
values  of  vl  and  v2  the  teeth  of  these  gears  must  be  so  formed 
as  to  permit  the  sliding  action,  and  to  transmit  a  velocity  ratio 
corresponding  to  the  rolling  action.  If  the  teeth  are  of  uniform 
cross-section,  and  the  pitch  elements  are  helices  making  angles 
a  and  /?  with  the  elements  of  the  respective  pitch  cylinders,  the 
sliding  may  take  place.  The  form  of  tooth  outlines  necessary 
to  transmit  constant  velocity  ratio  equivalent  to  pure  rolling  of 
the  normal  helices  is  determined  by  the  curvature  of  these  helices, 
being  the  same  as  that  for  spur  gears,  the  pitch  lines  of  which  have 
radii  equal  to  the  radii  of  curvature  of  the  respective  helices. 


152 


KINEMATICS  OF  MACHINERY. 


Such  gears  are  called  helical  gears.  They  are  included  (together 
with  worm  gears,  which  are  treated  in  the  next  chapter)  under 
the  general  head  of  screw  gears  in  the  classification  on  page  110. 
Helical  gears  are  also  commonly  known  as  spiral  gears.  They 
closely  resemble  in  form  the  twisted  gears  described  in  Article 
75,  but  their  action  is  entirely  different.  It  is  to  be  noted 
that  the  axes  of  twisted  spur  gears  are  always  parallel,  while 
helical  gears  may  be  designed  for  any  angle  between  the  axes, 
although  they  are  usually  at  right  angles.  The  tooth  action 
of  helical  gears  is  widely  different  from  that  of  twisted  gears. 
The  former  have  point  contact,  and  the  latter  line  contact, 
while  the  screw-like  action  of  helical  gears  results  in  a  large 
amount  of  sliding  in  the  direction  of  the  common  tangent  to 
the  tooth  elements.  This  sliding  action  is  entirely  absent  in 
the  case  of  twisted  gears.  In  twisted  gears  the  angular 
velocity  ratio  is  inversely  proportional  to  the  radii;  while  in 
helical  gears  it  depends  not  only  on  the  radii,  but  also  on  the 
relative  values  of  the  angles  a  and  /?. 

83.  The  Pitch  of  Helical  Gear  Teeth.— Fig.  130  represents 
the  pitch  surface  of  a  helical  gear.  The  pitch  elements  of  the 

teeth  are  shown  crossing  the  cylin- 
der elements  at  an  angle  (f>,  called 
the  angle  of  cut  of  the  teeth.  The 
pitch  p  of  these  teeth  is  the  dis- 
tance between  corresponding  pitch 
elements  of  adjacent  teeth  measured 
along  a  normal  helix.  The  dis- 
tance between  the  same  elements 
measured  on  a  transverse  section 
of  the  pitch  cylinder  is  p  -=-  cos  (/>.  If 
there  are  n  teeth,  xd=nXp •*•  cos  (p 


\ 


Fig.  130 


or  d=n-r-—  cos  6.     Since  —  =  p'  —  the  diametral  pitch  correspond- 
P  P 


ing  to  p,  the  equation  may  be  written 


'  cos  (j>,  or  n  —  d 


OUTLINES  OF  GEAR-TEETH.  153 

Xp'  cos  (/>.  The  corresponding  values  for  spur  gears  are  d  =  n 
+  p',  and  n  =  dXp'. 

84.  The  Velocity  Ratio  of  Helical  Gears. — If  dt  and  d2  are 
the  respective  diameters  of  A  and  B  in  Fig.  129  the  corresponding 

angular  velocities  are  (tjl  =  vl^-^  and  w2  =  v2+-^.  The  angular 
velocity  ratio  is  ~  =  ~Lj--  Since  i\  cos  a  =  v2  cos/?  this  equa- 

W2        V2dl 

a).     d2  cos  /? 

tion  may  be  written  —  =  3 .     It  is  evident  from  the  rela- 

a>2     dl  cos  a 

tion  between  the  diametral  pitch,  the  angle  of  cut  and  the  num- 
ber of  teeth  determined  in  the  preceding  article,  that  the  number 
of  teeth  of  A  is  n^  =  d^p'  cos  a  and  the  number  of  teeth  of  B  is 

n2=d2pf  cos  /?.      Substituting    in   the    above    equation,    •- 1  =  — , 

(o2     n^ 

from  which  it  appears  that  the  angular  velocity  ratio  of  a  pair 
of  helical  gears  is  equal  to  the  inverse  ratio  of  the  number  of 
teeth  of  the  respective  gears.  This  relation  is  the  same  as  for 
spur  and  bevel  gears,  and  all  other  classes  of  gears. 

85.  Outlines  of  Helical  Gear  Teeth. — It  was  stated  in  Art. 
82  that,  for  constant  angular  velocity  ratio,  the  tooth  outlines 
of  helical  gears  must  have  the  same  shape  as  those  of  spur  gears 
having  radii  equal  to  the  radii  of  curvature  of  the  normal  helices. 
The  radius  of  curvature  of  a  helix  crossing  the  elements  of  a  cylinder 
at  any  angle  is  equal  to  that  at  the  end  of  the  minor  axis  of  the 
ellipse  in  which  a  plane  making  the  same  angle  with  the  axis  of  the 
cylinder  cuts  the  surface.    In  Fig.  131,  d  is  the  diameter  of  the 
pitch  cylinder  of  a  helical  gear;  sos  is  a  tooth  element;  non  is 
a  normal  helix;  </>  is  the  angle  of  cut.     The  major  axis  of  the 
ellipse  cut  from  the  cylinder  by  a  plane  making  the  angle  <j>  with 
a  transverse  section  of  the  cylinder  is  d~cos  <f>.     The  radius  of 

curvature  at  the  end  of  the  minor  axis  is  (- 7)   ^-= — - 

\2  cos  $/       2     2  cos2  (/> 

Hence  the  tooth  outlines  of  the  helical  gear  should  be  the  same 


154 


KINEMATICS  OF  MACHINERY. 


as  those  of  a  spur  gear  the  diameter  of  which  is  d-r-cos2  <j>.     The 
number  of  teeth  of  a  helical  gear  is  n  =d  p'  cos  <£,  that  of  a  spur 

d       .  ^ !_ 

n 


gear  of  diameter  d  +  cos2  </>  is  ri  = 


,  from 


cos*  9*  n      cosa 

which  it  appears  that  the  teeth  of  a  helical  gear  having  n  teeth 


Fig.  131 

should  have  the  same  outlines  as  those  of  a  spur  gear  having 
n-7-cos3  <j)  teeth. 

86.  Graphical  Method  for  Helical  Gears.  —  The  relations 
between  the  dimensions  of  any  pair  of  helical  gears  may  be. 
shown  graphically  as  in  FigT  132.  The  angle  between  the  axes  = 
xoy=0.  xy=dl  +  d2  =  2c=-  twice  the  distance  between  shaft 
centres;  angle  oxy  =  90°  —  a,  and  angle  oyx=9Q°—  /?;  gx  =  dt 
and  gy  =  d2;  gh  and  gk  are  perpendicular  to  ox  and  oy  respectively^ 
gb  is  parallel  to  ox,  and  ga  is  parallel  to  oy. 

The  following  relations  are  evident: 


Angle  /i(/a=angle  kgb;  .'.  gh  :gk  ::ga  :gb. 

It  has  been  shown  (Art.  83)  that  d1  =  n1-^-pf  cos  af  and  d2= 
n±  -f-  p'  cos  /?.  In  Fig.  132,  d1  -•=  gh  -+-  cos  a,  and  d2  =  gk  cos  /?.  Equat- 
ing the  respective  values  of  d±  and  d2, 

gh  =  nv  -T-  pf,     and     gk  =  n2  +  pf. 

From  these  equations  it  appears  that  gh  and  gk  are  respectively 
equal  to  the  pitch  diameters  of  a  pair  of  spur  gears,  each  of  which 


OUTLINES   OF  GEAR-TEETH. 


155 


has  the  same  pitch  and  number  of  teeth  as  the  corresponding 
helical  gear.     Combining  these  equations, 

gh_ni_«>2.  QhjE.J!b.      -    ^i=^ 

gk~n2~col'  gk     gb     oa'       '  n>2     ob 


Also  xy=2c=dl  +  d2  =  nl  +  p'  cosa+n2  +  p'cosp.     Since  /?=#-«, 
this  equation  may  be  written  nx  -r-  cos  a  -f  n2  -f-  (cos  0  —  a)  =  2p'c. 


Fig.  132 

A  similar  diagram  (Fig.  133)  may  be  constructed  when  only 
the  centre  distance,  c;  the  shaft  angle,  6;  the  velocity  ratio, 
col  -=-  oj2  •  and  the  diametral  pitch,  p',  are  given.  From  this  diagram 
the  values  of  nt  and  n2,  d^  and  d2J  and  a  and  /?  may  be  determined. 

Draw  ox  and  oi/  making  the  angle  6  at  o.  Lay  off  oa  and  06 
on  ox  and  oi/,  respectively,  so  that  oa  :ob  1:0^  :  a)2. 

Draw  06  and  6e  parallel  to  oy  and  ox  respectively,  intersecting 
at  e.  Draw  oe.  Draw  the  line  XT/,  equal  in  length  to  2c,  inter- 
secting oe  at  #,  so  that  the  distance  og  is  a  maximum.  Draw  #/i 
and  gk  perpendicular  to  ox  and  oy  respectively.  Multiply  the 
lengths  of  gh  and  gk  by  the  diametral  pitch,  p't  to  obtain  approxi- 
mate values  of  nv  and  w2.  For  the  actual  values  of  nt  and  n2 
take  the  largest  whole  numbers,  not  greater  than  the  approximate 


156 


KINEMATICS  OF  MACHINERY. 


values,  wJiich  will  satisfy  the  equation  n2  +  nl  =  a)l  +  a)2.  Compute 
the  corresponding  values  of  n^p'  and  n2+p',  and  take  h'g' 
and  k'g'  equal  to  these  values  and  parallel  to  hg  and  kg  respectively, 
locating  gf  on  oe.  Through  g'  draw  x'y'  equal  in  length  to  xy, 
terminating  in  ox  and  oy  at  x'  and  y'  respectively.  Then  angle 
x'g'h'  =  a,  and  angle  y'g'kf  =  p;  distance  g'x'  =  dv  and  g'y'  =  d2. 
Since  it  is  not  possible  to  obtain  exact  values  by  construction, 
these  values  should  be  considered  as  approximate.  The  exact 


Fig.  133 


value  of  a  is  obtained  by  trial  from  the  equation  ^-r-co 
4- cos  (8— a)  =  2p'c,  using  the  approximate  value  first  and 
changing  it  slightly  until  an  exact  equality  results.  The  corre- 
sponding exact  value  of  /?=#—  a;  those  of  dt  and  d2  are  obtained 
from  the  equations  dl  =  nl-i-pf  cos  a,  and  d2  =  n2-±p'  cos  /?. 

When  the  shafts  are  at  right  angles,  as  is  usually  the  case, 
0  =  90°,  and  cos  (0  —  a)  =  sin  a.  Substituting  this  value,  the 
above  equation  becomes  n^cos  a  +  n2-v-sin  a  =  2p'c.  This  may 
be  written  n±  tan  a  +  n2=2p'c  sin  a. 


OUTLINES  OF  GEAR-TEETH.  157 

It  will  frequently  be  found  that  some  of  the  values  determined 
by  this  construction  are  not  suitable  for  actual  gears.  The  next 
less  number  of  teeth  in  each  gear  that  will  give  the  required 
velocity  ratio  may  then  be  tried.  If  this  does  not  give  satis- 
factory values,  either  the  pitch  or  the  distance  between  centres 
must  be  changed., 

87.  Cast  Gears. — Gear-teeth  are  either  cut  in  a  machine 
or  are  cast.  For  the  rougher  classes  of  work,  it  is  common 
practice  to  use  gears  with  cast  teeth;  but  cut  gears  are  now  used 
almost  exclusively  for  the  better  grades  of  work.  When  gears 
are  cast,  it  is  impoitant  to  form  the  patterns  very  carefully, 
and  especially  to  space  the  teeth  accurately.  With  the  utmost 
care,  however,  it  is  impossible  to  get  very  smooth  and  accurately 
spaced  teeth,  so  clearance  between  the  sides  of  the  teeth,  or 
backlash,  must  be  provided.  With  small  gears  the  enlargement 
of  the  mould,  due  to  "  rapping  "  the  pattern,  more  than  com- 
pensates for  the  shrinkage,  and  unless  this  is  looked  after  in  the 
pattern  shop  and  foundry,  the  teeth  may  be  too  thick  when  cast. 

A  convenient  device  for  forming  the  teeth  of  the  pattern  is 
shown  in  Fig.  134.  A  block  of  hardwood  (preferably  of  a  color 
quite  distinct  from  the  wood  of 
the  pattern)  is  shaped  as  shown, 
so  that  the  sections  at  amr  and 
AMR  correspond  to  the  two 
ends  of  the  teeth  (for  a  spur- 
gear  these  sections  are,  of  course,  p-tg  I34 
alike).  The  middle  portion  is 

cut  out,  so  that  the  distance  L  equals  the  length  of  a  tooth; 
that  is,  the  part  removed  corresponds  to  the  form  of  a  tooth. 
The  stock  for  the  teeth  of  the  pattern  is  gotten  out  in  lengths 
equal  to  L,  and  large  enough  in  cross-sections  to  make  a  tooth. 
The  block  of  Fig.  134  is  screwed  in  the  vise,  and  it  may  have 
two  pointed  brads  projecting  upward  through  the  bottom. 
Then  a  piece  of  the  prepared  stock  is  forced  down  into  the  space 


158  KINEMATICS  OF  MACHINERY. 

in  this  "  form/'  and  is  then  planed  up  with  "  hollow  and  round  " 
planes.  By  working  down  to  the  form  it  is  quite  easy  to  produce 
a  large  number  of  teeth  very  uniform  in  shape. 

Where  many  large  cast  gears  are  made,  a  gear  moulding 
machine  is  sometimes  used,  as  it  produces  accurate  work  and 
reduces  the  cost  of  patterns.  A  stake,  or  arbor,  is  set  upright  at 
the  centre  of  the  mould,  which  may  be  swept  up  in  loam  to 
approximately  the  outside  form  of  the  wheel.  The  pattern  for  the 
teeth,  simply  a  block  with  a  few  teeth  attached  which  corresponds 
to  a  segment  of  the  entire  rim,  is  fastened  to  the  stake  by  an  arm. 
This  arm  holds  the  segmental  pattern  at  the  proper  distance  from 
the  axis  of  the  wheel,  and  the  arm  and  pattern  can  be  turned  about 
this  axis.  The  pattern  can  also  be  withdrawn  towards  the  centre 
or  upward.  In  moulding  the  gear,  this  segment  is  placed  in 
position  and  a  few  teeth  are  moulded  by  filling  in  about  the  pattern 
with  the  sand.  The  pattern  is  then  drawn,  rotated-  about  the  axis 
through  a  small  angle  (one  or  two  teeth  less  than  the  number  in 
the  pattern),  and  a  few  more  teeth  are  moulded.  In  this  way  the 
entire  rim  is  moulded  by  sections.  An  index  plate,  or  ring,  is  used 
to  insure  accurate  spacing.  After  completion  of  the  rim  the 
pattern  (or  the  cores)  for  the  arms,  hub,  etc.,  is  used  to  complete 
the  mould. 

88.  Methods  of  Cutting  Gear-teeth. — \7hen  a  gear  having 
cut  teeth  is  to  be  made  a  gear-blank  is  prepared  which  is  identical 
with  the  finished  gear  in  every  respect,  except  that  it  has  no 
teeth.  The  teeth  are  then  formed  in  this  blank  by  cutting  out 
the  spaces  between  them.  This  may  be  done  either  by  planing 
or  milling.  In  planing  machines  the  tool  has  a  reciprocating 
motion,  and  cuts  during  the  stroke  in  one  direction  only;  in 
milling  machines  rotary  cutters  are  used;  and  the  cutting  is 
continuous  during  the  process  of  forming  a  space.  In  Fig.  135, 
a  illustrates  a  tool  used  for  planing  gear-teeth,  and  b  shows  a 
standard  milling-cutter.  The  cutting  edges  of  both  of  these 
tools  are  formed  to  the  exact  shape  of  the  space  between  two 


OUTLINES  OF  GEAR-TEETH. 


159 


teeth  of  the  gear  to  be  cut.     Another  class  of  tools  have  cutting 

edges  formed  to  the  shape  of  the  tooth  outline  of  another  gear 

of  the  same  pitch  as  the  one  to  be  cut. 

An  important  difference  between  these 

two  classes  of  tools  is  that  while  the 

range  of  the  number  of  teeth  in  a  gear 

that   may  be  cut  with  a  single  cutter 

of  the  first   class  is  very  narrow,  any 

gear  from  the  smallest  pinion  to  a  rack 

may  be  cut  with  a  single  cutter  of  the 

second  type.     On  the  other  hand,  it 

should  be  noted  that  the  former  class 

of  tools  may  be  used  in  the  ordinary 

milling  and  shaping  machines,  without  Fifl<  135t 

any  special  attachments  other  than  an 

index-head  for  rotating  the  blank  into  the  correct  positions   for 

cutting  the  teeth,  while  the  latter  class  is  used,  in  general,  only 

in  machines  designed  especially  for  cutting  gears. 

To  reduce  the  wear  of  the  finishing  tool,  it  is  customary 
(except  in  fine  pitch  gears)  to  roughly  form  the  teeth  by  "gashing  " 
the  blank  before  the  final  cutting  operation.  When  the  teeth 
of  a  gear  have  been  thus  roughed  out,  they  may  be  finished  by 
planing  with  a  tool  ground  to  a  sharp  point  which  cuts  the  tooth 
surfaces  element  by  element. 

In  the  following  articles  the  use  of  these  methods  in  Cutting 
spur,  bevel,  twisted  and  helical  gears  will  be  briefly  outlined. 

A  detailed  description  of  these  operations  and  of  the  machines 
which  perform  them  may  be  found  in  a  book  entitled  "  Gear 
Cutting  Machinery  "  by  Mr.  R.  E.  Flanders. 

89.  Planing  Spur-gears. — When  the  teeth  of  a  spur-gear 
are  to  be  cut  with  a  tool  such  as  that  shown  in  Fig.  135a,  the 
gear-blank  is  mounted  between  the  centres  of  an  indexing  mechan- 
ism, placed  on  the  table  of  a  planing  or  shaping  machine.  The 
stroke  of  the  tool  is  parallel  to  the  axis  of  the  gear.  Between 


160 


KINEMATICS  OF  MACHINERY. 


strokes  the  tool  is  fed  radially  toward  the  axis  of  the  blank  until 
the  required  depth  of  space  is  obtained.  The  tool  is  then  with- 
drawn, and  the  indexing  mechanism  used  to  turn  the  blank  on 
its  axis  into  the  proper  position  for  cutting  the  next  space.  This 
process  is  repeated  until  all  the  teeth  are  completed.  This  is  the 
simplest  method  of  cutting  gear  teeth.  It  may  be  used  for  any 
system  of  tooth  outlines.  It  is  especially  adapted  to  cutting 
the  teeth  of  annular  gears,  and  gears  of  large  diameter.  It  is 
also  useful  when  a  gear  is  needed  and  no  standard  milling-cutter 
is  available. 

A  hardened  steel  pinion  (properly  "  backed  off  "  to  give  cutting 
clearance)  having  a  reciprocating  motion  parallel  to  its  axis  may 
be  used  as  a  cutter  for  spur-gears.  The  gear-blank  is  mounted 
with  its  axis  parallel  to  that  of  the  cutter.  The  cutter  and  the 
blank  are  connected  by  a  train  of  mechanism  so  that  as  the  cutter 
is  rotated  very  slightly  on  its  axis  between  strokes,  the  blank 
also  turns  on  its  axis  with  a  velocity  ratio  corresponding  to  pure 
rolling  contact  between  the  pitch-cylinders.  The  cutter  thus 
meshes  with  the  gear  it  is  cutting  as  shown  in  Fig.  136.  At  the 
beginning  of  the  operation  of  cutting  a  gear  by  this  method  the 
cutter  is  fed  toward  the  blank  until  the  distance  between  the  axes 

corresponds  to  tangency  of  the 
pitch-cylinders  of  the  cutter 
and  the  blank.  The  rotation 
on  the  axes  is  then  begun.  All 
the  teeth  are  completed  when 
one  rotation  of  the  blank  on  its 
axis  has  been  made.  A  single 
cutter  serves  to  cut  all  gears  of 
a  given  pitch. 

A  tool  having  a  cutting  edge 
shaped  like  a  single  tooth  of  a 
rack  of  the  same  pitch  as  the  gear  to  be  cut  may  also  be  used  in  a 
somewhat  similar  manner.  In  this  case  the  slight  rotations  of  the 


Fig.  136. 


OUTLINES   OF  GEAR-TEETH. 


161 


blank  on  its  axis  are  accompanied  by  a  motion  of  the  tool  in  a 
direction  at  right  angles  to  the  stroke,  the  relative  motion  corre- 
sponding to  pure  rolling  between  the  pitch  surface  of  the  gear 
being  cut  and  that  of  the  imaginary  rack  of  which  the  cutter  is  a 
tooth.  Fig.  137  shows  the  position  of  the  tool  relative  to  the 
space  being  cut  at  several  stages  during  the  operation.  By 
indexing  the  blank  after  every  stroke  of  the  cutter,  equal  cuts  are 
taken  on  all  of  the  teeth.  The  rolling  action  between  the  pitch 
surface  of  the  gear  and  that  of  the  imaginary  rack  is  independent 


Fifl.  137. 

of  the  indexing,  and  occurs  each  time  the  blank  has  been  indexed 
through  a  complete  revolution. 

Spur-gear  teeth  which  are  too  large  to  be  cut  with  formed  tools 
may  be  finished  with  a  planing  tool  having  a  sharp  point,  after  the 
spaces  have  been  roughly  cut  out  by  other  means.  In  this  opera- 
tion the  feeding  of  the  tool  toward  the  axis  of  the  blank  is  accom- 
panied by  a  lateral  movement  at  right  angles  to  the  feed,  causing 
the  cutting  point  to  trace  the  outline  of  the  tooth.  In  this  way 
the  teeth  are  formed  element  by  element.  The  lateral  movement 
is  usually  produced  by  a  templet  formed  to  the  exact  shape  of  the 
tooth  outline.  After  one  side  of  one  tooth  has  been  completed 
the  blank  is  indexed  and  the  operation  repeated  on  the  next 
tooth.  By  providing  a  second  tool  to  work  simultaneously 
on  the  other  side  of  the  teeth,  the  necessity  for  turning  the 
blank  over  after  all  the  teeth  have  been  finished,  on  one  side  b 
avoided. 


162 


KINEMATICS  OF  MACHINERY. 


90.  Milling  Spur-gears. — When  a  standard  milling-cutter, 
such  as  is  shown  in  Fig.  1356,  is  to  be  used  to  cut  spur-gear 
teeth,  the  blank  is  mounted  with  its  axis  at  right  angles  to  the 
axis  of  rotation  of  the  cutter.  The  blank  is  fed  toward  the  cutter 
and  the  whole  space  between  two  teeth  is  cut  out  by  one  passage 
of  the  cutter  across  the  face  of  the  blank.  The  blank  is  then 
turned  into  position  for  cutting  the  next  space,  and  the  operation 
repeated  until  all  the  teeth  are  completed. 

Standard  cutters  are  made  in  sets  suitable  for  cutting  all 
gears  of  a  given  pitch  from  a  12-tooth  pinion  to  a  rack.  The 
following  table  shows  the  cutters  in  one  set  for  epicycloidal  and 
involute  teeth,  as  manufactured  by  the  Brown  &  Sharpe  Mfg.  Co. 


EPICYCLOIDAL  SYSTEM 

INVOLUTE  SYSTEM 

24  Cutters  in  each  Set. 

12  Cutters  in  each  Set. 

Cutter. 

Teeth 

Cutter. 

Teeth. 

Cutter. 

Teeth. 

A  cuts 

12 

M  cuts 

27  to    29 

1  cuts 

135  to  rack 

B 

13 

N 

30  to    33 

2 

55  to  134 

C 

14 

O 

34  to    37 

3 

35  to    54 

D 

15 

P 

38  to    42 

4 

26  to    34 

E 

16 

Q 

43  to    49 

5 

21  to    25 

F 

17 

R 

50  to    59 

6 

17  to    20 

G 

18 

s 

60  to    74 

7 

14  to    16 

H 

19 

T 

75  to    99 

8 

12  to    13 

I 

20 

U 

100  to  149 

J 

21  to  22 

V 

150  to  249 

K 

23  to  24 

w 

250  or  more 

L 

25  to  26 

X 

rack 

For  absolute  accuracy  a  different  cutter  would  be  required 
for  each  number  of  teeth  of  each  pitch.  Practically  this  is  not 
necessary,  as  the  change  in  the  form  of  the  teeth  for  a  small 
change  in  number  is  slight,  except  in  case  of  gears  having  few 
teeth.  The  form  of  teeth  changes  more  rapidly  for  epicycloidal 
than  for  involute  teeth.  For  this  reason  a  larger  number  of  cut- 
ters is  required  in  an  epicycloidal  set. 

When  a  large  number  of  duplicate  spur-gears  are  to  be  pro- 
duced, the  teeth  are  usually  milled  with  a  hob  similar  to  that 


OUTLINES  OF  GEAR-TEETH. 


163 


shown  in  Fig.  154a.  When  the  helix  angle  of  the  thread  of  the 
hob  is  <fi,  the  hob  is  mounted  with  its  axis  of  rotation  making 
the  angle  90°  —  ^  with  the  axis  of  the  gear-blank.  The  blank 
and  the  hob  are  connected  by  a  train  of  gears  so  that  as  the  hob 
rotates  the  blank  turns  on  its  axis.  The  velocity  ratio  of  the 
hob  and  blank  is  equal  to  the  number  of  teeth  of  the  gear  to  be 
cut,  divided  by  the  number  of  threads  of  the  hob.  The  hob  is 
fed  slowly  in  a  direction  parallel  to  the  axis  of  the  blank,  and 


Fig.  138. 


Fig.  138  ; 


as  it  cuts  across  the  face  of  the  blank  the  spaces  between  the 
teeth  are  cut  to  their  full  depth,  all  the  teeth  being  completed 
by  a  single  passage  of  the  hob  across  the  face  of  the  blank.  Fig. 
138  shows  the  relative  positions  of  the  hob  and  blank  in  an  early 
stage  of  the  cutting.  Teeth  cut  by  this  method  are  not  theo- 
retically accurate  on  account  of  the  interference  between  the 
threads  of  the  hob  and  the  teeth  of  the  gear.  This  interference 
is  due  to  the  fact  that  the  helix  angle  of  the  thread  elements 
of  the  hob  is  less  than  $  when  these  elements  are  outside  the 
pitch-surface,  and  greater  than  $  when  they  are  inside  the  pitch- 
surface.  In  Fig.  138a,  the  pitch  element  of  a  hob-thread  is 
represented  by  ss,  while  sY  and  s"s"  are  thread  elements  respect- 


164  KINEMATICS  OF  MACHINERY. 

ively  inside  and  outside  of  the -pitch-surf  ace.  The  corresponding 
tooth-elements  of  a  spur-gear  being  cut  by  the  hob  are  1 t,  ft' 
and  t"t"j  all  of  which  are  parallel.  While  the  pitch  elements 
ss  and  1 1  are  tangent  at  o,  s's'  and  ft'  intersect  at  of  and  s"s'r 
and  t"t"  intersect  at  o".  This  interference,  which  is  greatly 
exaggerated  in  the  figure,  tends  to  undercut  the  flanks  and  to 
cut  away  the  outer  ends  of  the  teeth.  The  use  of  a  single-thread 
hob  of  comparatively  large  diameter  reduces  the  interference 
to  a  minimum. 

91.  Cutting  Twisted,  Helical,  and  Bevel-gears. — If  the  motion 
of  the  tool  across  the  face  of  the  gear-blank  is  accompanied 
by  a  uniform  rotation  of  the  blank  on  its  axis  the  elements  of 
the  teeth  cut  will  be  helical.  The  cutter  axis  must,  of  course, 
be  set  at  the  proper  angle  with  the  gear  axis.  In  this  way  helical 
and  twisted  gears  may  be  cut,  using  any  of  the  methods  described 
for  spur-gears,  the  turning  of  the  blank  on  its  axis  being  produced 
by  a  suitable  train  of  gears.  The  tooth  outlines  of  gears  pro- 
duced in  this  way  will,  in  general,  not  be  theoretically  correct. 
Twisted  gears  of  correct  tooth  outline  may  be  planed  with  a 
cutting  edge  formed  to  the  shape  of  the  space  between  the  teeth 
of  a  spur-gear  of  the  same  pitch  and  diameter.  The  tool  must, 
however,  be  formed  to  work  in  a  helical  groove.  Accurate  helical 
gears  may  be  cut  by  planing  with  a  tool  having  a  cutting  edge 
formed  to  the  shape  of  the  tooth  of  a  rack  of  the  same  pitch. 
The  method  usually  employed  for  helical  gears  is  either  milling 
with  a  standard  cutter  or  hobbing.  In  both  operations  there 
is  interference  between  the  cutter  and  teeth  of  correct  outline, 
so  that  the  resulting  teeth  are  not  theoretically  correct.  The 
error  is  so  small,  however,  that  it  can  be  neglected  in  most 
cases. 

It  has  been  shown  (Art.  78)  that  the  tooth  elements  of  a 
perfect  bevel-gear  all  converge  to  the  apex  of  the  pitch  cone.  It 
is  therefore  impossible  to  cut  a  bevel-gear  having  theoretically 
correct  tooth  outlines  by  any  method  using  a  tool  formed  to  the 


OUTLINES    OF    GEAR-TEETH.  165 

shape  of  the  space  between  two  teeth,  as  this  space  not  only 
changes  in  width  in  passing  across  the  face  of  the  gear,  but  the 
tooth  outlines,  though  similar,  grow  smaller  as  they  approach  the 
apex  of  the  pitch  cone.  Milling-cutters  are,  however,  much  used 
for  bevel-gears.  To  make  the  space  between  the  teeth  narrower 
at  one  end  than  at  the  other  it  is  necessary  to  take  at  least  two 
cuts  for  each  tooth  space.  By  shifting  the  axis  of  rotation  between 
these  cuts  it  is  possible  to  make  the  pitch  elements  of  the  teeth 
pass  through  the  apex  of  the  pitch  cone.  The  thickness  of  the 
cutter  used  should  not  exceed  the  width  of  the  space  at  the 
small  end;  the  outline  of  the  edge  usually  corresponds  to  that 
of  the  large  end  of  the  teeth.  The  resulting  teeth  are  too  thick 
above  the  pitch  line  at  the  small  end.  This  extra  thickness 
may  be  removed  by  filing.  Similar  results  are  obtained  when 
bevel-gear  teeth  are  planed  with  a  tool  formed  to  the  shape  of 
the  tooth  space.  This  method  should  be  used  only  for  bevel- 
gears  of  rather  narrow  face. 

Accurate  bevel-gear  teeth  may  be  cut  by  means  of  a  tool 
having  a  sharp  j)oint,  the  strokes  of  which  are  directed  toward 
the  apex  of  the  pitch  cone,  while  the  tooth  outlines  are 
determined  by  templet  guides,  or  some  equivalent  arrange- 
ment. This  method  can  be  used  only  when  the  teeth  have 
been  previously  roughed  out  in  the  blank.  By  using  two 
templets  and  two  cutting  points  both  sides  of  the  teeth  may 
be  formed  simultaneously. 

Another  method  that  produces  accurate  involute  bevel-gear 
teeth  is  illustrated  in  principle  in  Fig.  139.  This  is  a  modification 
of  the  process  of  planing  spur-gears  with  a  tool  having  an  edge 
formed  to  the  shape  of  the  tooth  o'f  an  involute  rack,  and  it  depends 
on  the  fact  that  the  tooth  outlines  of  an  involute  crown-gear  (a 
bevel-gear  the  pitch  elements  of  which  are  perpendicular  to  the 
axis)  are  identical  with  those  of  an  involute  rack  of  the  same 
pitch.  The  crown-gear  A  meshes  with  a  master-gear  B  having 
the  same  pitch  angle  as  the  gear  to  be  cut.  The  gear-blank  C 


166 


KINEMATICS  OF  MACHINERY. 


having  teeth  roughly  cut  is  carried  on  the  same  shaft  as  the 
master-gear,  and  is  rigidly  connected  to  the  master-gear,  except 
when  it  is  being  indexed.  The  pitch-cone  of  the  blank  is  tan- 
gent to  the  pitch-plane  of  the  crown-gear,  and  rolls  on  this  plane 
when  the  master-gear  and  crown-gear  turn  on  their  axes.  The 
tool  D  has  a  straight  cutting  edge,  corresponding  to  one  side  of 
an  involute  rack  tooth.  The  tool-slide  (not  shown  in  the 
figure)  moves  on  a  guide  attached  to  the  crown -gear  so  that 


Fig.  139. 

the  edge  of  the  tool  describes  one  side  of  the  tooth  of  an 
imaginary  crown-gear  meshing  with  the  gear  being  cut.  The 
motion  of  the  point  of  the  tool  is  toward  the  apex  of  the 
pitch  cone,  along  the  line  OE.  Between  strokes  of  the  tool 
the  master-gear  is  turned  through  a  slight  angle,  causing  the 
crown-gear  and  the  blank  to  turn  slightly  on  their  axes.  By  this 
means  a  pure  rolling  action  is  obtained  between  the  pitch  surfaces 
of  the  gear  being  cut  and  the  imaginary  crown-gear  whose  tooth 
is  described  by  the  edge  of  the  tool.  When  one  side  of  one  tooth 
has  been  completed  the  blank  is  indexed  on  its  axis  into  position 


OUTLINES   OF   GEAR-TEETH.  167 

for  forming  the  next  tooth.  Other  mechanisms,  giving  identical 
motion,  may  be  substituted  for  the  crown-gear  and  master-gear 
shown  in  Fig.  139. 

By  the  use  of  two  tools  both  sides  of  the  teeth  may  be  formed 
simultaneously.  The  blank  may  be  indexed  after  each  stroke 
of  the  tool,  instead  of  upon  completion  of  a  tooth.  In  this  case 
the  rolling  action  occurs  on  the  completion  of  each  rotation  of 
the  blank  by  the  indexing  mechanism. 

Bevel  gears  may  also  be  cut  with  a  special  hob.  used  in  a  special 
machine,  but  the  operation  is  too  complex  to  be  described  here. 

92.  Other  Classes  of  Gearing. — In  the  table  at  the  beginning 
of  this  chapter  six  classes  of  gearing  are  mentioned.  Examples 
of  all  these  classes  except  Skew  and  Face-gears  have  been  dis- 
cussed in  this  chapter.  Skew-gears  and  Skew  Bevel-gears  are 
those  based  on  rolling  hyperboloids  as  pitch  surfaces.  These 
rolling  hyperboloids  were  very  briefly  treated  in  Art.  53;  but 
the  theory  of  the  teeth  of  these  wheels  is  so  complex,  and  their 
application  is  so  very  rare,  that  a  discussion  of  them  is  hardly 
warranted  in  a  short  general  treatise. 

Face-gearing  is  an  almost  obsolete  class,  formerly  used  when 
wooden  gears  prevailed,  because  the  teeth  (mere  pegs  or  pins) 
were  easily  made.  The  consideration  of  this  class  will  also  be 
omitted. 

Twisted  Bevel  and  Skew  Gears  are  derived  from  the  corre- 
sponding bevel  and  skew-gears  in  the  same  way  that  twisted 
spur-gears  are  derived  from  the  ordinary  spur-gears. 

Worm-gears,  which  are  a  form  of  screw-gearing,  will  be  treated 
in  the  next  chapter. 

The  Practical  Treatise  on  Gearing,  published  by  the  Brown  & 
Sharpe  Mfg.  Co.,  gives  a  great  many  valuable  points  on  the 
actual  construction  of  gearing;  such  as  directions  for  laying  out 
blanks  for  cut  gears,  etc. 

Grant's  Teeth  of  Gears  is  a  most  excellent  concise  treatise  on 
tooth  outlines;  and  MacCoid's  Kinematics,  which  is  devoted 


163  KINEMATICS  OF  MACHINERY. 

mainly  to  gearing,  is  a  very  complete  work;  covering  many 
points  not  ordinarily  taken  up  and  containing  much  original 
matter. 

The  discussion  of  this  chapter  has  been  very  much  abbreviated, 
as  the  subject  is  exhaustively  treated  in  a  large  number  of  available 
books,  and  no  attempt  has  been  made  to  give  more  than  a  general 
treatment  of  fundamental  principles. 


CHAPTEE  V. 

CAMS  AND  OTHER  DIRECT-CONTACT  MECHANISMS. 

93.  Cams. — The  term  cam  is  applied  to  a  large  and  varied  class 
of  machine  members  which  are  often  used  to  impart  a  more  or  less 
complex  motion  to  a  follower.  Cams  are  most  commonly  employed 
for  motions  which  cannot  be  easily  produced  by  other  simple  forms 
of  mechanism.  A  cam  consists  of  a  piece  so  shaped  that  its  motion 
(which  is  usually  a  rotation,  but  often  an  oscillation  or  translation) 
imparts  a  definite,  and  ordinarily  variable,  motion  to  another  mem- 
ber upon  which  it  acts  by  direct  contact,  or  through  an  auxiliary 
roll  or  block. 

Figs.  36,  44,  and  57  show  common  forms  of  cams.  Such  mech- 
anisms as  those  illustrated  in  Figs.  38,  39,  etc.,  and  even  gear-teeth, 
might  be  treated  as  special  forms  of  cams :  but  it  is  more  convenient 
to  consider  them  by  themselves. 

There  are  two  principal  classes  of  cams :  those  with  a  curved 
edge  or  groove,  which  impart  motion  to  a  follower  moving  in  the 
plane  of  the  cam  motion,  and  those  which  cause  the  driver  to  move 
in  a  different  plane,  usually  perpendicular  to  the  plane  of  the  cam's 
motion.  The  latter  class  may  be  derived  from  the  former,  as  will 
appear  later.  Figs.  140  to  147  represent  cams  of  the  first  kind. 

94.  Cams  moving  the  Follower  in  the  Plane  of  the  Cam  by  a 
Point  or  Roller. — Fig.  140  represents  a  simple  case,  in  which  the 
path  of  the  follower  is  a  straight  line  passing  through  the  fixed 
centre  about  which  the  driver  rotates.  Suppose  0  to  be  the  fixed 
centre  of  the  cam,  PM  to  be  the  path  of  a  point  P  of  the  follower, 
and  that  P  is  to  have  positions  corresponding  to  0,  1,  £,  3,  etc. ;  for 

169 


170 


KINEMATICS  OF  MACHINERY. 


the  angular  motions  of  the  driver  o,  an  a2?  etc.  These  angles  maybe 
equal  or  otherwise ;  and  the  follower  may  have  a  period  of  rest,  or  a 
''dwell,"  as  indicated  by  the  coincidence  of  the  positions  4,  5.  Lay 
off  the  radii  01',  02',  03',  etc.,  making  the  desired  angles  a19  «2, 
etc.,  with  PO,  and  locate  the  corresponding  positions  of  the  fol- 
lower, 0,  1,  2,  3,  etc.  With  a  centre  at  0  and  radius  01,  draw  an 
arc  cutting  01'  at  1';  with  radius  02,  cut  02'  at  2',  etc.  Through 


Fig.  140. 


these  intersections  pass  a  smooth  curve.  Then  this  curve  as  it  ro- 
tates will  impart  the  required  motion  to  the  point  P,  which  is  sup- 
posed to  be  guided  in  the  line  PM,  for  when  the  cam  has  moved 
through  the  angle  <*, ,  for  example,  the  radius  01'  will  coincide  with 
the  line  PM,  and  1'  will  be  at  1.  The  same  reasoning  applies  to 
all  the  points  found  above. 

If  the  real  cam  is  a  solid  (cylinder)  of  which  the  curve  shown 
is  one  of  the  equal  transverse  sections,  the  follower  would  have 
to  be  a  mere  edge,  to  satisfy  the  conditions  given.  While  such  a 
combination  satisfies  the  kinematic  requirements,  it  would  work 
with  unnecessary  friction  and  would  wear  rapidly;  hence  a  derived 
form  is  used  which  gives  the  same  motion  to  the  follower. 


CAMS  AND  OTHER  DIRECT-CONTACT  MECHANISMS.     171 

To  reduce  the  friction,  a  roller  with  its  centre  at  P  may  be 
attached  to  the  follower,  as  shown  by  the  small  circle  in  the 
figure.  The  profile  of  a  cam  to  impart  the  required  motion  to 
the  follower  by  acting  on  this  roller  is  determined  by  the  follow- 
ing construction :  Take  a  radius  equal  to  the  radius  of  the  roller, 
and  with  centres  along  the  original  outline  draw  arcs  inside  that 
outline.  A  smooth  curve  tangent  to  all  these  arcs  is  the  required 
outline.  The  original  outline  is  called  the  pitch  line  of  the  derived 
cam. 

When  the  path  of  the  point  P  of  the  follower  is  in  a  straight 
line  which  does  not  pass  through  O,  or  when  it  is  curved,  as  in 


•--*'        Fig.  141. 


Fig.  141,  the  construction  differs  slightly  from  that  of  Fig.  140. 
The  radii  of  the  cam  are  laid  off  as  before  at  angles  corresponding 
to  the  required  successive  angular  motions  of  the  cam.  When 
the  radii  01',  02',  etc.,  lie  in  the  line  OS,  the  follower  centre,  P, 
is  not  on  these  radii,  but  in  the  path  PM,  at  1,  2,  etc.  With  0 
as  a  centre  and  a  radius  01,  draw  the  arc  1-1"-1"'-1'.  Now 
lay  off  on  this  arc,  from  its  intersection  with  the  radius  01',  a 
distance  equal  to  1-1",  locating  the  point  1"'.  This  is  a  point  in 
the  required  pitch  line,  for  when  I'  is  at  1",  the  point  V"  coincides 


172 


KINEMATICS  OF  MACHINERY. 


with  1.  The  other  points  in  the  pitch  line  are  located  in  a  similar 
way.  The  distances  I'-l'",  2'-2'",  etc.,  are  laid  off  to  the  right  or 
left  of  1',  2',  etc.,  according  as  the  positions  1,  2,  etc.,  of  the  fol- 
lower are  to  the  right  or  left  of  OS. 

The  actual  cam  outline  to  act  properly  with  a  roll  of  any  given 
diameter  is  obtained  precisely  as  in  the  other  example. 

It  is  usually  desirable  to  have  the  path  of  the  follower  as  nearly 
in  line  with  a  radius  of  the  cam  as  possible,  as  this  condition  gives 
less  obliquity  of  action,  especially  with  small  cams. 

95.  Cams  acting  on  a  Tangential  Follower  to  move  it  in  the 
Plane  of  the  Cam. — It  is  often  desired  to  have  the  cam  act  tan- 


M 

415 


Fig.  142. 


gentially  upon  a  flat  or  a  curved  follower,  as  in  Figs.  142,  143,  or 
144.  In  this  case  there  are  limitations  to  the  motion  which  it  is 
possible  to  impart  to  the  follower.  Thus  in  Figs.  142  and  143,  in 
which  the  acting  surface  of  the  follower  is  a  flat  face  (plane  sur- 
face), it  is  evident  that  no  portion  of  the  acting  surface  of  the 
cam  can  be  concave,  for  such  portions  could  not  become  tangent 
to  the  follower. 

With  the  curved  (convex)  follower  as  shown  in  Fig.  144,  it  is 


CAMS  AND   OTHER  DIRECT-CONTACT  MECHANISMS.     173 

possible  to  have  a  concave  portion  of  the  driver,  but  this  portion 
must  have  as  great  a  radius  of  curvature  as  any  part  of  the  follower 
with  which  it  acts.  General  methods  of  designing  these  tangential 
cams  are  shown  in  Figs.  142,  143,  and  144. 

In  Fig.  142  the  various  positions  of  the  follower  are  parallel  to 
each  other,  and  the  acting  face  is  preferably,  but  not  necessarily, 
perpendicular  to  /the  direction  of  the  follower's  motion.  Lay  off 
radii  of  the  cam,  01',  02',  etc.,  marking  desired  angular  motions 
from  the  original  position  corresponding  to  the  given  positions  of 
the  follower,  1-1,  2-2,  etc.  With  a  centre  at  0,  draw  arcs  through 
the  intersections  of  the  various  positions  of  the  follower  with  the 
reference  line  OM,  and  cutting  the  corresponding  radii  of  the  cam, 
as  shown,  at  1',  2',  etc.  At  1',  2',  3',  etc.,  draw  lines  making  the  same 
angle  with  the  respective  radii  that  the  follower  makes  with  PM. 
Draw  a  smooth  curve  tangent  to  all  these  lines  last  drawn.*  This 
curve  is  the  required  cam  outline;  for  when  any  radius,  as  0  3', 
lies  on  OM,  3'  will  lie  at  3  and  the  tangent  through  3'  coincides 
with  the  required  position  of  the  follower. 

If  the  successive  positions  of  the  follower  are  not  parallel  to 
each  other,  as  in  Fig.  143,  the  solution  is  as  follows: 

3 


Fig.  143. 

With  a  centre  at  0,  draw  arcs  through  the  intersections  of  O'-l, 
0'  -3,  etc.,  with  the  reference  line  OAf ;  cutting  the  respective 

*  If  it  is  found  impossible  to  draw  a  curve  tangent  to  all  these  lines,  a 
condition  as  to  the  successive  positions  of  driver  and  follower  has  been  impose  1 
which  cannot  be  met  with  this  type  of  cam.  If  the  curve  has  a  sharp  corner 
or  crosses  itself,  the  difficulty  can  usually  be  overcome  'by  increasing  the 
diameter  of  the  cam. 


174 


KINEMATICS  OF   MACHINERY. 


radii  01',  02',  etc.,  at  I',  2',  etc.  At  1',  draw  a  line  making  an 
angle  with  the  radius  01 '  equal  to  fa;  at  2'  draw  a  line  making  an 
angle  with  the  radius  02'  equal  to  fa,  etc.,  then  proceed  as  in  the 
former  example  by  drawing  a  curve  tangent  to  these  last  lines. 

When  the  follower  is  curved  and  has  an  angular  motion  as  in 
Fig.  144,  the  following  modification  of  the  above  method  may 

be  used.  Connect  0'  with  the  points 
1,  2,  etc.,  where  the  successive  posi- 
tions of  the  curved  edge  of  the  fol- 
lower cut  OM.  Draw  circular  arcs 
about  0  with  radii  01,  02,  etc., 
cutting  the  radii  01',  02',  etc., 
which  correspond  to  the  desired 
angular  motions  of  the  cam.  Draw 
a  circle  through  0'  with  centre  at  0; 
then  with  radii  O'l,  0'2,  etc.,  and  centres  at  1',  2',  etc.,  cut  this 
last  circle  in  the  points  1",  2",  etc.  Now  form  a  templet  with 
the  curve  of  the  follower  for  one  edge,  and  a  centre  corresponding 
to  0'.  Place  this  centre  at  1",  with  the  edge  of  the  templet 
passing  through  1',  and  trace  along  the  edge.  In  a  similar  way  with 
the  templet  centre  at  2",  and  the  edge  passing  through  2',  trace 
the  curve;  and  so  on  for  all  the  phases  required.  A  smooth  curve 
tangent  to  all  these  traced  positions  of  the  follower  is  the  required 
cam  outline;  for  when  2"  rotates  with  the  cam  to  0',  2'  will  fall 
at  2,  and  the  corresponding  tracing  of  the  templet  will  coincide 
with  the  required  position  of  the  follower,  and  hence  the  cam  will 
be  tangent  to  this  position  of  the  follower.  Similar  reasoning 
applies  to  the  other  phases. 

96.  Positive  Return  of  Follower. — In  the  forms  of  cams  con- 
sidered so  far,  no  means  has  been  shown  for  insuring  a  return 
of  the  follower  after  it  reaches  its  position  farthest  from  the  centre 
of  the  cam.  This  is  often  accomplished  through  the  action  of 
gravity,  a  spring,  or  some  other  external  force;  but  it  is  necessary 
under  many  conditions  to  completely  constrain  the  motion  of 


CAMS  AND  OTHER  DIRECT-CONTACT  MECHANISMS.     'J75 

the  follower  by  the  mechanism  itself.  The  follower  is  then 
said  to  have  a  positive  return.  Positive  return  of  the  fol- 
lower is  sometimes  obtained  by  providing  a  groove  in  which 
the  roll  of  the  follower  acts  as  in  Fig.  145.  In  this  arrange- 
ment there  are  two  defects,  which  may  be  serious,  especially 
at  high  speeds.  The  slot  must 
have  a  width  somewhat  in  ex- 
cess of  the  diameter  of  the  roll, 
for  both  faces  of  the  cam  groove 
move  in  the  same  direction, 
and  if  the  roll  is  in  contact 
with  the  inner  face  of  the  cam 
it  tends  to  rotate  in  one  direc- 
tion, while  if  it  touches  the  outer  \^  "~~~  *^/Flg<  14S* 
face  this  tends  to  rotate  the 
roll  in  the  opposite  direction.  The  figure  shows  the  action 
when  the  inner  face  is  acting  on  the  roll.  Owing  to  the  necessary 
clearance  there  is  some  lost  motion  which  is  taken  up  when  the 
follower  reaches  either  of  its  extreme  positions,  and  is  acted  upon 
by  the  other  face  of  the  cam.  If  the  speed  is  high  the  taking 
up  of  this  "  slack  "  may  result  in  a  sharp  blow  or  knock  which 
makes  a  noise  and  may  injure  the  mechanism.  The  other  defect 
is  due  to  the  fact  that  when  the  roll  changes  contact  from  one 
face  to  the  other,  there  is  a  tendency  to  instant  reversal  of  its 
motion,  but  the  inertia  of  the  rapidly  revolving  roll  resists  this, 
resulting  in  a  temporary  grinding  action  which  wears  both  cam  and 
roll.  Under  slow  speeds  these  actions  may  not  be  serious. 

A  method  of  returning  the  follower  which  overcomes  these 
defects  is  shown  in  Fig.  146.  Two  rolls  on  opposite  sides  of  the 
cam  shaft  are  mounted  as  shown,  or  in  some  similar  manner.  A 
cam  is  designed,  by  the  method  given  in  Art.  94,  to  impart  the 
required  motion  to  one  of  these  rolls.  Every  position  of  this 
roll  causes  the  other  to  occupy  a  definite  position,  due  to  the 
connection  between  them,  and  a  complimentary  cam  is  designed 


176 


KINEMATICS  OF  MACHINERY. 


146. 


corresponding  to  these  various  positions  of  the  second  roll.  This  cam 
is  placed  beside  the  first  one  on  the  same  shaft,  and  its  action  on  the 

second  roll  keeps  the  first  roll  in 
contact  with  its  cam,  and  imparts 
the  return  notion  to  the  follower. 
A  single  cam  will  give  a  pos- 
itive return  motion  to  a  sliding 
follower  having  two  rolls  of  equal 
diameter  in  contact  with  the  cam 
on  opposite  sides  of  the  centre. 
The  rolls  must  be  so  mounted 
that  a  line  joining  their  centres 
will  pass  through  the  centre  of 
the  shaft.  With  this  construction  the  return  motion  of  the  fol- 
lower will  be  an  exact  duplicate  of  the  forward  motion. 

A  complementary  cam  may  also  be  used  to  secure  positive 
return  of  the  follower  with  cams  of  the  types  shown  in 
Figs.  142,  143,  and  144.  When  the  forward  and  return  motions 

are   alike  a   yoke  follower  of  the 

type  shown  in  Fig.  147  may  be 
used  with  a  single  cam. 

A  cam  like  the  one  shown  in 
Fig.  147  can  be  designed  very 
easily,  as  it  is  bounded  by  circular 
arcs.  The  follower  is  shown  in  its 
extreme  position  to  the  right. 
There  is  a  "  dwell,"  or  period  of 
rest,  at  both  extreme  positions  of 
the  follower.  The  parts  ab  and  cd 
are  arcs  with  a  common  centre  0, 
and  the  sum  of  their  radii  equals  the  distance  between  the  parallel 
working  faces  of  the  follower,  =D.  To  draw  the  arcs  be  and  ad 
take  a  radius  equal  to  D,  and  draw  an  arc  be  with  a  centre  on  )(fc?. 
This  arc  must  be  tangent  to  ab  at  b.  Now  with  the  same  radius, 


Fig.  147. 


CAMS  AND  OTHER    DIRECT-CONTACT  MECHANISMS.    177 

Dj  and  a  centre  at  c  draw  the  arc  ad,  thus  completing  the  out- 
line. In  the  position  shown  the  follower  is  at  rest;  when  a  comes 
in  contact  with  the  left-hand  face  of  the  follower,  on  the  line  of 
centres,  c  is  in  contact  with  the  other  face  at  a  point  directly 
opposite;  then  while  ad  acts  upon  the  left-hand  face  of  the 
yoke  the  follower  moves  to  the  left;  this  is  followed  by  a  period 
of  rest  while  dc  is  in  contact  with  the  left-hand  face,  and  then 
ad  comes  in  contact  with  the  right-hand  face,  returning  the  fol- 
lower to  the  right.  This  cam,  or  a  modification  of  it,  is  used. to 
actuate  the  valves  of  the  engines  on  the  stern-wheel  steamers  of 
the  upper  Mississippi  and  its  tributaries. 

97.  Translation  Cams. — A  form  of  cam  which  by  its  translation 
imparts  motion  to  a  follower  is  shown  in  Fig.  148.   If  it  be  required 


5"    6 


that  a  point  of  the  follower  shall  be  at  the  points  1,  2,  3,  etc., 
as  the  points  1',  2',  3',  etc.,  of  the  driver  coincide  with  0',  design 
the  cam  as  follows:  Erect  perpendiculars  at  1',  2',  etc.,  as  I'-l", 
etc.  Diaw  a  line  from  1,  parallel  to  the  motion  of  the  driver, 
cutting  I'-l"  in  1";  through  2  draw  a  parallel,  cutting  2'-2" 
in  2",  etc.  A  line  through  these  successive  intersections  gives 
the  pitch  line  of  the  cam.  If  0,  1,  2,  represent  positions  of  the 
centre  of  a  roll  attached  to  the  follower,  the  actual  cam  outline 
is  formed  as  indicated  in  Fig.  140. 

If  the  successive  motions  of  the  follower  are  to  be  proportional 
to  those  of  the  driver,  the  cam  becomes  an  inclined  plane,  as 
shown  in  Fig.  149.  In  this  case  the  flat  shoe  as  shown  may  act 
directly  on  the  cam,  as  it  will  fit  the  surface  of  the  driver  at  all 
points.  This  will  provide  a  larger  contact  surface  and  reduce 
wear,  though  it  results  in  greater  friction  than  when  a  roll  is  used. 


178 


KINEMATICS   OF   MACHINERY. 


98.  Motion  imparted  to  the  Follower  Perpendicular  to  Plane  of 
Cam's  Motion. — The  cam  of  either  Fig.  148  or  149  may  be  wrapped 
upon  a  right  cylinder,  as  shown  in  Figs  150  and  151,  and  then  the 
rotation  of  these  cylinders  about  their  axes  will  impart  a  motion 
to  the  follower  parallel  to  these  axes  and  exactly  equivalent  to  the 
motion  due  to  the  translation  of  the  original  cams.  The  base 
lines,  or  lines  parallel  to  the  translation  of  the  original  cams, 
become  circles  when  the  cams  are  wrapped  upon  the  cylinders. 

If  these  cams  are  provided  with  grooves  in  which  a  roll  acts,  as 
in  Fig.  150,  clearance  must  be  provided:  for  the  opposite  faces  of 
the  cam  surface  tend  to  rotate  the  roll  in  different  directions; 
hence  these  cams  are  subject  to  the  defects  of  the  grooved  cam  men- 


Fig.  150. 


-<b-"\          Fig.  151. 

L _\ 

tioned  in  a  preceding  article.  There  is  another  peculiarity  of 
the  action  of  such  a  earn  as  that  of  Fig.  150,  if  the  roll  is  a  cylinder, 
which  results  in  a  grinding  action  between  the  cam  face  and  the 
roll;  but  this  is  overcome  by  using  a  properly  proportioned  conical 
roll,  as  shown  in  the  figure.  If  the  outer  radius  of  the  cam  is  rt  and 
the  radius  at  the  bottom  of  the  working  part  of  the  groove  is  ra, 
points  at  the  outer  edge  have  a  linear  velocity  of  ^nr^n,  while 
points  at  the  inner  portion  have  a  smaller  velocity,  %7rr9n.-  If 
the  roll  is  a  cylinder,  the  faces  of  the  cam  being  perpendicular  to 
the  axis,  all  points  on  the  contact  element  of  this  cylinder  must 
evidently  have  equal  linear  velocities;  hence  the  velocities  of  con- 
tact points  of  the  driver  and  follower  can  only  be  the  same  at  one 
ooint  along  the  element  of  contact.  If,  however,  the  roll  is  the 
,  ustum  of  a  cone  with  its  apex  at  the  axis  of  the  cam  and  the 


CAMS  AND  OTHER  DIRECT-CONTACT  MECHANISMS.     179 


sides  of  the  groove  be  given  the  corresponding  slant,  this  difficulty 
is  practically  overcome.     The  shaded  portion  of  the  face  in  the 
section  at  the  right  of  Fig.  150  shows  the  development  of  the  act- 
ing surface  of  such  a  conical  roll,  or  of  a  portion  of  such  surface. 
If  the  cam  is  to  rotate  continuously,  instead  of  vibrating  upon  its 
axis,  the  original  cam  of  Fig.  148  must  have  its  first  and  last  ordinates 
of  equal  lengths  (measured  from  the  base  line,  or  from  any  line  par- 
allel to  this) ;  otherwise  the  groove  would  not  be  continuous  when 
formed  on  the  cylinder,  and  it  could  only  drive  the  cam  by  recipro- 
cation about  its  axis ;  when  the  re- 
turn motion  would  be  an  exact  re- 
versal of  the  forward  motion.    The 
path  of  the  follower  may  be  other 
than  a  straight  line,  and  the  con- 
struction of  the  pitch  line  of  such  a 
cam  is  shown  in  Fig.  152.  The  points 
in  the  pitch  line  are  on  the  parallels 
to  the  path   of  the  driver;  but  to 
the  right,  or  left,  -of  the   intersections  of  such  parallels  with  the 
corresponding    perpendiculars    by   distances   equal    to   the    given 
simultaneous  distance  of  the  follower  from  the  reference  line  0-M. 
The  construction  of  the  cam  from  this  pitch  line  is  exactly  similar 
to  the  cases  already  treated. 

99.  The  Screw. — The  cam  of  Fig.  151,  as  derived  from  the  in- 
clined plane  of  Fig.  149,  will  be  recognized  as  the  common  screw, 
in  which  the  sliding  block  of  the  follower  corresponds  to  a  portion 
of  the  thread  of  the  nut. 

The  ordinary  nut  and  screw  differs  essentially  from  this  only  in 
the  length  of  the  block  of  the  follower,  which  is  made  to  include 
several  coils  of  the  cam  (threads  of  the  screw)  in  order  to  distribute 
the  pressure,  reduce  wear,  and  increase  the  strength.  The  groove 
in  which  the  nut  works  may  be  rectangular,  triangular,  or  of  any 
one  of  many  possible  sections,  without  modifying  the  relative  motion 
transmitted,  which  is  governed  entirely  by  the  relation  between  the 
axial  and  circumferential  components  of  the  threads. 


180  KINEMATICS   OF  MACHINERY. 

If  the  cam  of  Fig.  149  be  wrapped  upon  a  cylinder  the  circum- 
ference of  which  is  L=?rZ),  each  revolution  of  the  cam  or  screw 
will  move  the  follower  through  a  distance  h,  which  is  in  this  case 
the  pitch  of  the  helix  or  screw-thread.  If  this  cam  were  wrapped 
upon  a  cylinder  of  half  the  above  diameter,  two  revolutions  of  the 
screw  would  be  required  to  move  the  follower  the  distance  h,  or 
the  cam  of  length  L,  as  shown,  would  make  two  complete  coils  of 
the  helix,  and  the  pitch  of  the  screw  (the  distance  from  one  thread 
to  the  next,  parallel  to  the  axis)  would  be  $h.  In  any  such  case 
the  inclination  of  the  helix  to 'the  plane  of  motion  of  any  point  in 
it  is  the  angle,  <£,  whose  tangent  is  /i-f-L,  and  this  inclination 
determines  the  velocity  ratio  of  driver  to  follower,  which  is  L  -f-  h, 
at  a  contact  point  on  the  pitch  line.  Of  course  the  groove  has  sen- 
sible depth  in  an  actual  screw,  and  then  points  on  the  screw-thread 
at  different  distances  from  the  axis  have  different  linear  velocities 
relative  to  the  nut.  If  the  screw  is  driven  by  a  crank  or  pulley  of 
larger  radius  than  the  acting  surface,  the  velocity  of  the  actual 
point  of  application  of  the  driving  force  is  correspondingly  greater, 
relative  to  the  nut,  than  L  ~-  h,  but  is  still  proportional  to  this 
quotient. 

If  the  pitch  of  the  thread  is  great  enough  to  permit  it,  another 
similar  thread  may  be  cut  between  the  grooves,  as  indicated  by  the 
dotted  lines  on  Fig.  151,  and  corresponding  extra  threads  of  the 
nut  may  work  in  this  groove.  This  does  not  alter  the  motion  trans- 
mitted, but  it  gives  more  bearing  surface  for  a  given  length  of  nut. 
Such  a  screw  is  called  a  double-threaded  screw.  There  may  be  any 
number  of  such  threads,  if  the  proportions  will  permit,  giving  a 
multiple-threaded  screw. 

100.  The  Endless  Screw.  Worm-gearing. — Fig.153  shows  a  screw 
with  an  angular  thread  and  a  small  block  (indicated  by  the  shaded 
tooth)  for  a  follower.  Rotation  of  the  screw  in  the  direction  indi- 
cated moves  this  follower  to  the  left.  If  this  block  is  pivoted  at  0' 
instead  of  being  guided  parallel  to  the  axis  of  the  screw,  it  will 
move  in  a  circle  about  0'  (provided  it  is  relieved  so  as  to  avoid 
binding),  and  it  soon  passes  out  of  action.  Now  if  a  series  of  such 


CAMS  AND  OTHER  DIRECT-CONTACT  MECHANISMS.     181 

pieces  be  arranged  in  a  continuous  circle  about  0',  all  connected 
together  and  properly  spaced,  as  each  piece  passes  out  of  action  at 
the  left  another  will  engage  at  the  right.  If  these  blocks  consti- 
tute a  complete  circular  rim,  the  action  will  be  continuous,  and 
indefinite  rotation  of  the  driver  will  result  in  indefinite  rotation  of 
the  follower.  This  arrangement  constitutes,  in  a  rudimentary 
manner,  the  common  worm  and  wheel,  or  endless  screw,  as  it  is 
sometimes  termed.  If  the  teeth  of  the  wheel  in  this  figure  were 
given  a  transverse  section  exactly  fitting  the  spaces  between  the 


Fig.  153. 


threads  of  the  screw,  as  is  the  case  in  an  ordinary  nut  and  screw, 
the  teeth  would  bind  or  interfere;  but  by  giving  a  properly  modi- 
fied form  to  the  teeth  the  action  may  be  made  smooth  and  uniform. 
Suppose  that  the  screw,  or  worm,  as  it  is  commonly  called,  be 
translated  axially  without  rotation.  It  will  then  cause  the  wheel  to 
rotate  just  as  a  pinion  is  rotated  by  a  rack ;  and  if  sections  of 
the  teeth  of  the  wheel  and  the  screw-threads  perpendicular  to 
the  axis  of  the  wheel  are  correct  forms  for  teeth  of  a  pinion  and 
rack,  the  velocity  ratio  will  be  constant.  It  will  be  apparent 
that  the  rotation  of  the  screw  is  equivalent  to  this  translation 
of  the  screw  when  acting  as  a  rack ;  for  by  this  rotation  succes- 
sive equal  meridian  sections  of  the  worm  are  brought  into  action 
on  the  middle  section  of  the  wheel.  It  follows  from  this  that  these 


182  KINEMATICS  OF  MACHINERY. 

sections  should  correspond  to  the  forms  proper  for  a  rack  and 
pinion.  The  sections  of  the  teeth  of  an  involute  rack  are  trape- 
zoids  (sections  of  the  acting  faces  being  inclined  straight  lines 
perpendicular  to  the  line  of  action,  or  the  common  normal). 
Then  if  the  screw-threads  of  Fig.  153  have  meridian  sections 
bounded  by  inclined  straight  lines  on  the  acting  sides,  the 
transverse  section  of  the  teeth  should  be  corresponding 
involutes.  The  detail  at  the  right  of  the  figure  indicates  this 
form. 

It  is  evident  that  if  the  worm  of  Fig.  153  is  a  single-threaded 
screw,  each  revolution  of  the  worm  causes  the  wheel  to  rotate 
through  an  arc  equal  to  the  pitch  arc  of  the  teeth ;  and  to  make  a. 
complete  revolution  of  the  wheel  the  worm  must  be  given  as  many 
turns  as  there  are  teeth  on  the  wheel.  The  pitch  angle,  as  in  spur- 
gears,  is  the  angle  between  two  teeth.  If  the  worm  is  a  double- 
threaded  screw,  the  helical  pitch  or  lead  is  twice  the  distance  from 
one  thread  to  the  corresponding  point  on  the  next  thread;  and 
one  revolution  of  the  worm  moves  two  teeth  of  the  wheel  past 
the  line  of  centres.  In  this  case  the  number  of  turns  of  the 
worm  required  to  produce  one  revolution  of  the  wheel  is  equal 
to  the  number  of  teeth  of  the  wheel  divided  by  two. 

In  general,  the  ratio  of  the  angular  velocity  of  the  worm  to. 
that  of  the  wheel  equals  the  number  of  teeth  of  the  wheel 
divided  by  the  number  of  separate  threads  of  the  screw.  The 
screw  is  called  a  worm  only  when  it  has  but  few  threads.  The 
number  of  threads  may  be  increased  indefinitely  with  a  cor- 
responding reduction  of  the  velocity  ratio.  It  is  customary  to 
design  the  teeth  of  the  resulting  gears  in  an  entirely  different- 
manner,  which  has  already  been  explained  in  the  articles  on 
helical  gears. 

In  designing  a  worm  and  wheel,  let  T=  number  of  threads 
(teeth)  on  wheel;  t= number  of  threads  on  worm;  p  =  circular 
pitch  of  wheel  =  axial  pitch  of  worm-threads;  D  =  pitch  diameter 
of  wheel;  d= pitch  diameter  of  worm;  J  =  distance  between  axes; 


CAMS  AND  OTHER  DIRECT-CONTACT  MECHANISMS.      183 

w  and  a>i  =  angular  velocities  of  wheel  and  woim,  respectively; 
9  =  helix  angle,  or  inclination  of  threads  to  transverse  section  of 
the  worm. 

OL-L.          •    T-^f 
~~ 


This  relation  usually  fixes  T  and  t. 

Tp  =  nD]  D=Tp  +  x;  or,  p  =  nD+T.  The  strength  of 
the  gear  depends  upon  p}  and  hence  p  should  be  fixed  and  D  made 
to  agree,  if  the  conditions  will  permit.  If,  however,  A  is  fixed,  Z> 
is  limited,  for  \  (D  +  d)  =  A  ;  but  D  and  d  may  have  any  value 
consistent  with  this  requirement.  The  value  of  p  gives  the  num- 
ber of  threads  to  the  inch  on  the  worm,  and  hence  p,  d  and  t  give 
the  helix  angle  <j>;  for  tan  (j>  =  pt+r.d. 

If  the  teeth  of  the  wheel  are  ordinary  screw-threads  (all  trans- 
verse sections  of  the  wheel  being  identical  in  form)  upon  a 
cylindrical  pitch  surface,  this  pitch  cylinder  and  that  of  the  worm 
are  tangent  at  a  single  point,  and  the  teeth  have  point  contact  only. 
That  is,  the  worm  always  engages  with  points  on  the  central 
transverse  section  of  the  wheel.  The  worm-wheel  may  be  made  of 
the  form  shown  in  Fig.  154,  when  it  is  called  a  close-fitting  wheel. 
The  teeth  of  this  wheel  may  be  drawn  by  passing  a  series  of  planes 
through  the  worm,  parallel  to  thefaxis  and  to  each  other  and  per- 
pendicular to  the  axis  of  the  wheel.  Each  of  these  sections  of  the 
worm  will  be  a  rack  section,  but  they  are  not  all  alike.  Then 
make  the  corresponding  sections  of  the  wheel  those  appropriate 
for  wheels  to  work  with  such  racks.*  This  process  is  tedious,  and  is 
seldom  required  in  practice,  as  by  the  method  of  cutting  the 
wheels  it  is  not  necessary  to  lay  out  the  teeth.  If  a  cast  worm  and 
wheel  are  to  be  made,  it  is  of  course  necessary  to  lay  out  the 
teeth. 


*  See  Unwin's  Machine  Design,  Part  I,  Art.  234,  for  a  full  description  of 
this  method  of  drawing  worm-wheel  teeth. 


184 


KINEMATICS   OF    MACHINERY. 


101.  Hobbing  Worm-wheels. — A  worm-wheel  maybe  accurately 
cut  by  the  following  process :  Turn  up  the  blank  to  correspond  to 
the  outside  of  the  teeth  (Fig.  154)*.  Next  cut  a  screw  of  tool  steel 
to  the  exact  form  of  the  worm,  then  make  a  milling-cutter  of  this 
tool  steel  worm  by  cutting  flutes  across  the  threads,  and  "  backing 


Fig.  154. 


off  "  the  teeth  thus  formed  for  clearance.  This  is  called  a  "  hob  " 
(see  Fig.  154a),  and  it  is  hardened,  tempered,  and  then  used  as  a 
milling-cutter.  The  hob  and  worm-wheel  blank  are  mounted  in 
the  gear-cutting  machine,  with  their  axes  at  right  angles  but 
necessarily  somewhat  farther  apart  than  the  desired  distance 

*  Figs.  154,  154a,  and  1546  are  taken  from  Brown  &  Sharpe's  Treatise 
on  Gears. 


CAMS  Ai\D  OTHER  DIRECT-CONTACT  MECHANISMS.     185 

between  the  axes  of  the  worm  and  wheel.  They  are  then  rotated 
about  their  axes  with  the  velocity  ratio  that  the  worm  and  wheel 
are  to  have,  and  the  blank  is  fed  toward  the  hob  very  slowly  until 
the  distance  between  the  axes  is  the  same  as  the  desired  distance 
between  the  axes  of  worm  and  wheel.  The  wheel  is  sometimes 
caused  to  rotate  simply  by  the  driving  action  of  the  hob,  the  teeth 
of  the  wheel  having  been  roughly  cut  or  "gashed"  with  an  ordinary 
milling  cutter ;  but  better  results  are  attained  when  it  is  driven 
positively  from  the  cutter-spindle,  with  the  required  velocity  ratio, 
through  a  suitable  train  of  gearing.  It  will  be  seen  that  the  teeth 
thus  formed  on  the  wheel  will  work  correctly  with  a  worm  which 
is  an  exact  reproduction  of  the  hob,  except  that  the  cutting-teeth 
are  omitted. 

The  worm  and  hob  may  be  cut  like  any  screw  in  a  lathe,  with  a 
tool  which  will  give  the  desired  form  of  threads. 

Fig.  1546  shows  a  method  of  cutting  an  approximate  close- 
fitting  worm-wheel  with  an  ordinary  gear-cutter,  the  diameter  and 
section  of  which  corresponds  to  the  worm.  If  the  cutter,  as  shown 
in  this  figure,  is  fed  diagonally  across  the  wheel-blank,  a  straight 
(point  contact)  wheel  will  be  produced. 

If  the  cutter  is  fed  radially  inward,  toward  the  axis  of  the 
wheel,  a  "drop-cut"  worm-wheel  is  produced.  Such  a  drop-cut 
wheel  resembles  a  bobbed  wheel  in  form  ;  but  the  method  does  not 
give  a  truly  close-fitting  wheel,  such  as  is  obtained  by  the  hobbing 
process. 


CHAPTEE  VI. 
LINKWORK. 

102.  General  Scope  of  Linkwork. — The  simplest  form  of  a  con- 
strained link-mechanism  consists  of  four  links,  each  pivoted  at  two 
points  to  adjacent  links.  A  link  with  hut  two  pivots,  and  joined 
to  two  adjacent  members,  is  called  a  simple  link.  If  a  link  has 
more  than  two  such  pivots  and  is  joined  directly  by  them  to  more 
than  two  separate  members  it  is  called  a  compound  link. 

A  complete  linkwork  "chain,"  as  link-mechanisms  are  some- 
times called,  cannot  have  less  than  four  links ;  for  if  three  links  are 
connected  in  a  closed  chain  they  form  a  triangle,  which  is  a  rigid 
construction  not  permitting  relative  motion  between  the  members. 
If  more  than  four  simple  links  are  connected  in  a  closed  chain,, 
forming  a  jointed  polygon  of  more  than  four  sides,  a  given  motion 
of  one  member  does  not  compel  the  others  to  move  in  a  definite 
manner.  Link-mechanisms  of  more  than  four  members  are  used  ;, 
but,  in  these  cases,  one  or  more  of  the  members  must  be  a  compound 
link.  Linkwork  can  be  used  to  convert : 

(a)  Continuous  rotation  into  reciprocation  (rectilinear  or  circu- 
lar) or  the  reverse. 

(b)  Reciprocation  into  reciprocation  with  a  constant  or  a  vari- 
able angular  velocity  ratio. 

(c)  Continuous  rotation  into  continuous  rotation,  with  a  COD- 
stant  or  a  variable  velocity  ratio. 

One  or  more  of  the  links  in  a  linkwork  chain  may  be  replaced 
by  a  sliding  block  or  similar  piece.  Certain  of  these  modified 
chains  are  of  very  great  importance  in  practical  machine  con- 
struction. 

186 


LIXKWORK. 


187 


103.  The  Four-link  Chain. — The  general  form  of  the  four-link 
chain  is  shown  by  Figs.  50  to  53  and  other  figures  already  given. 
It  is  now  in  order  to  examine  the  influence  of  the  proportions  of 
the  members  of  the  four-link  chain  upon  the  motion  transmitted. 

The  following  notation  will  be  used  :  The  driver  will  be  desig- 
nated as  a  ;  the  follower,  b ;  the  connector,  c\  and  the  stationary 
link,  or  frame,  d. 

Fig.  155  shows  a  mechanism  in  which  circular  reciprocation  of 
a  produces  circular  reciprocation  of  b.  The  phase  shown  by  the 
light  lines  a',  b',  c' ,  is  a  limiting  phase,  and  a  can  move  no  farther 
to  the  left  (left-hand  rotation).* 

The  driver  (Fig.  155)  can  move  through  the  arc  a'-a-a"-a'"9 
causing  the  follower  to  move  from  V  through  b  to  5",  and  then 
return  over  this  path  to  b'".  Within  these  limits  circular  recipro- 
cation can  produce  circular  reciprocation,  and  either  member  might 
be  the  driver;  but,  practically,  the  action  is  not  smooth  when  the 


follower  is  near  a  dead-point;  and  if  b  is  the  follower,  the  range  of 
action  should  be  somewhat  less  than  the  maximum  given.  In  this 
figure  it  will  be  seen  that  b-\-c<a-\-d:,  and  it  will  be  seen  that 

*  This  position  of  the  follower,  &',  is  called  a  dead-point  position;  for  there 
can  be  no  component  of  the  motion  of  the  connector  in  the  direction  of  its  length, 
and  hence  no  positive  transmission  of  motion.  When  the  connector  and  either 
the  driver  or  follower  lie  in  one  straight  line,  a  dead-point  is  reached.  If  tl  e 
two  members  lie  on  opposite  sides  of  their  common  pivot,  as  6'  and  c'  in  Fig. 
155,  the  condition  is  called  an  outer  dead-point  position.  If  the  links  coincide 
in  direction  and  are  on  the  same  side  of  their  pivot,  as  b'  and^c'  in  Fig.  156,  an 
inner  dead-point  is  reached. 


188 


KINEMATICS  OF  MACHINERY. 


a  cannot  make  a  complete  rotation  unless  I  -f  c  is  equal  to,  or 
greater  than,  a  -f-  d\  or  c  —  a  >  d  —  b. 

In  Fig.  156  the  follower  reaches  an  inner  dead-point  when  it  is 
in  the  position  b' ',  and  a  can  rotate  no  farther  to  the  right  than  the 
position  a'.  In  this  case  the  driver  can  vibrate  through  the  angle 
af—a—a"-arf',  causing  the  follower  to  reciprocate  from  V  through 
b  to  b"  and  back  to  b'".  It  is  evident  that  d  <  than  a  -f-  c  —  b; 
and  the  driver  cannot  make  a  complete  rotation  unless  d  is  equal 
to,  or  greater  than,  a  -\-  c  —  b  \  or  a  -\-  c  <  d  -f  b.  It  will  be 
noticed  that  the  follower  might  pass  (but  cannot  be  positively 
driven  by  a)  beyond  the  position  #',  in  either  Fig.  155  or  Fig.  156; 
if  this  should  occur  in  any  way,  the  motion  transmitted  would  be 
completely  changed. 

To  sum  up  these  two  cases,  we  find  that  the  driver  cannot 
make  a  complete  revolution  unless  these  conditions  are  present: 
c  —  a  _>  d  —  b\  and  c  +  a  <^d  -\-  b.  li  c  —  a  =  d  —  b,  the  driver 
and  follower  have  simultaneously  inner  and  outer  dead-points, 
respectively,  as  shown  by  Fig.  157  (in  the  phase  #',  Z>'),  and  the 


c— a  =  d— 


:   „?  \^\d[    * 

\  J^^\ysN\N^sXX 


motion  of  the  follower  may  be  towards  either  b"  or  b'",  as  the 
driver  passes  this  position.  If  c  +  a  =  d  +  b  (Fig.  158)  the 
driver  reaches  the  outer  dead-point  as  the  follower  reaches  the 
inner  dead-point  (#'  and  b',  respectively);  and  the  follower  may 
either  return  to  b"  or  pass  on  to  b'".  If  c  —  a  >  d  —  b,  and 


LINKWORK. 


189 


c-f  a  <  d+6,  as  in  Fig.  159,  the  motion  of  the  follower  is  fully 
constrained,  and  the  driver  can  make  a  complete  rotation. 


/g\\\Nv^NN^^^J^\N^N^^cs^^\^^^^^^^^^c 
J      Fig.  159. 


104.  Continuous  Rotation  of  both  Driver  and  Follower. — (See 
Fig.  160.)  If  a  single  four-link  chain  is  used  to  transmit  positive 
continuous  rotation  to  a  follower  from  a  rotating  driver  there  must  be 
no  dead-points  ;  for  if  the  driver,  a,  reaches  a  dead-point,  b  will 
come  to  rest;  and  if  b  reaches  a  dead-point,  its  motion  will  not  be 
fully  constrained,  and  it  will  generally  lock  the  driver,  preventing 
complete  rotation  of  the  latter.  With  the  proportions  of  Fig.  160, 
neither  a  nor  b  reaches  a  dead-point,  and  either  of  these  members 
may  be  used  as  a  driver,  compelling  the  other  to  rotate  continuously, 
but  the  velocity  ratio  will  be  variable.  If  rotation  of  both  a  and  b 
is  to  be  continuous,  they  will  have  simultaneous  dead-points  if  either 
reaches  a  dead-point.  If  c  =  mn  =  d-}-b  —  a,a  will  have  an  outer 
dead-point,  and  b  will  have  an  inner  dead-point  at  the  same  instant. 
If  c  =  mn'  =  a  -\-  b  —  d,  a  and  b  will  both  have  inner  dead-points 
at  one  phase.  Hence,  for  continuous  positive  rotation  of  both  a 
and  b  the  following  conditions  must  be  fulfilled  :  c  >  d  +  b  —  «, 
and  c  <  a  +  b  —  d\  hence,  d  -\-b  —  a  <  a  -\-b  —  d  .'.  d  <  a.* 

The  mechanism  of  Fig.  160  is  called  a  "drag-link";  and  it  is 
sometimes  used  to  connect  the  two  arms  of  a  centre-crank  or  double- 
throw  crank.  In  this  case  a  and  b  are  equal,  and  d  equals  zero  in 
the  proper  adjustment,  that  is,  the  fixed  axes  of  a  and  b  coincide. 

*  If  dead-points  are  permissible  (as  in  the  parallel  rods  of  locomotives), 
other  provision  is  made  for  insuring  that  the  dead-points  shall  be  passed;  theu 
d  may  be,  and  usually  is,  greater  than  a. 


190  KINEMATICS  OF  MACHINERY. 

As  long  as  this  condition  is  maintained,  the  link  forms  the  equiva- 
lent of  a  rigid  connection,  and  as  the  mechanism  is  reduced  to  a 
three-link  chain,  the  motion  transmitted  is  exactly  similar  to  that 
of  a  solid  crank.  If  either  axis  is  shifted,  through  improper  align- 
ment, springing  of  the  shaft  supports,  or  wear,  the  motion  is  trans- 
mitted from  one  section  of  the  shaft  to  the  other  with  a  slight 
variation  in  their  angular  velocity  ratio  during  the  revolution,  and 
the  wrenching  action  on  the  shaft  is  much  less  than  it  would  be 
with  the  usual  form  of  rigid  crank-shaft. 

If  a  and  b  are  equal  the  angular  velocity  ratio  is  constant  when 
d  equals  zero,  or  when  d  =;  c\  for  with  these  proportions  the  two 
perpendiculars  from  the  fixed  centres  to  the  line  of  the  link  (c)  are 
always  equal  for  any  phase.  The  former  condition  (d  =  0)  is  that 
of  the  drag-link  as  applied  to  engine-cranks  in  proper  alignment. 
The  second  condition  (d  =  c)  is  one  met  in  the  locomotive  side-rod 
connection;  but  in  this  case  the  driver  and  follower  have  simulta- 
neous dead-points,  and  special  means  must  be  resorted  to  for  com- 
plete constrainment  of  the  follower. 

The  essentials  of  the  locomotive  side-rod  connection  are  shown 
in  Fig.  59,  in  which  0  and  0'  correspond  to  the  centres  of  the 
connected  wheels,  A'  B'  is  the  side  rod  (the  dotted  circles  represent 
the  paths  of  the  pins  by  which  the  side  rod  is  pivoted  to  the  wheels); 
OA'  and  O'E'  (radii  of  the  pin  circles)  are  the  driver  and  follower 
between  which  it  is  desired  to  transmit  rotation  with  a  constant 
velocity  ratio;  and  00'  (the  frame)  is  the  fourth  link.  The  full 
lines  of  Fig.  59  show  a  phase  at  which  the  driver  and  follower  both 
lie  on  the  line  of  centres.  As  the  driver  passes  its  dead-point  po- 
sition, the  follower  might  move  in  either  of  the  directions  indicated 
by  the  arrows  at  B.  Means  of  overcoming  this  defect  in  the  con- 
strainment will  be  shown  later. 

In  the  mechanism  under  consideration  it  is  necessary  that  the 
four  links  shall  form  a  parallelogram  in  all  phases;  that  is,  in  Fig. 
59,  A'B'  must  equal  00' \  and  OA'  must  equal  O'B'.  When  this 
condition  is  fulfilled  the  angular  velocity  ratio  must  always  be  unity, 
for  the  perpendiculars  from  the  fixed  centres  (0,  0')  to  the  con- 


LINKWORK. 


191 


nector  (A'B')  are  equal  in  any  phase  (see  Art.  30).  In  order  to 
insure  continuous  rotation  of  the  follower  when  the  dead-points  are 
passed,  the  simple  mechanism  of  Fig.  59  must  be  supplemented. 
The  method  used  on  locomotives  is  shown  in  Fig.  161. 

Each  axle  has  two  driving-wheels  secured  to  it;  the  two  wheels 
on  either  side  being  coupled  by  a  side  rod.     The  pins  on  the  two 


1 

1 

I 

1 

1 
1 
1 

1 

1 

1 

I 

1 

S" 

\ 

1 

1 

1 

__  c 


V 


c 


Fig.  162. 

wheels  of  each  axle  are  placed  so  that  they  are  not  in  line,  but  one 
of  these  pins  is  ahead  of  the  other,  as  shown  in  Fig.  161  by  the 
angle  0,00,=  0,'0'C,'. 

This  angle  is  commonly  90°,  so  that  when  the  system  is  at  a 
dead  centre  phase  on  one  side,  the  complementary  system  on  the 
other  side  is  in  the  best  phase  for  transmission  of  motion. 

Other  possible  arrangements  to  secure  complete  constrainment 
are  shown  in  Fig.  162.  In  this  case  three  equal  cranks,  not  neces- 
sarily having  their  centres  in  one  straight  line,  are  connected  by  a 
rigid  member  (a  compound  link)  which  has  a  bearing  for  the  pin 
of  each.  These  bearings  must  be  spaced  to  agree  with  the  spacing 
of  the  fixed  centres,  and  the  cranks  are  always  parallel  to  one  an- 
other. The  middle  crank  (shown  with  the  arrow)  should  be  the 
driver. 

105.  Combined  Linkwork  and  Sliding-block  Mechanisms. — The 
preceding  articles  of  this  chapter  have  been  devoted  to  the  four- 
link  chain,  and  it  was  seen  that  by  the  mechanism  of  Fig.  159  a 


192  KINEMATICS  OF  MACHINERY. 

circular  reciprocation  of  the  follower  may  be  imparted  by  the  re- 
ciprocation or  rotation  of  the  driver.  Fig.  163  shows  a  mechanism 
in  which  one  member  of  the  four-link  chain  is  replaced  by  a  curved 
block  bf  sliding  in  a  corresponding  circular  arc  groove  in  an  exten- 
sion of  the  fixed  link  d.  It  is  evident  that  the  motion  transmitted 


to  this  block  is  exactly  equivalent  to  the  motion  which  it  would  re- 
ceive if  it  were  connected  to  d  by  the  dotted  link  b'.  Whatever 
the  length  of  V,  it  may  be  replaced  by  a  block  and  groove,  the 
centre  line  of  which  corresponds  to  the  path  of  the  moving  end  of 
the  link  b',  without  altering  the  character  of  the  motion.  Any 
other  link  might  be  replaced  in  a  similar  way  by  the  equivalent  slot 
and  block.  When  any  one  of  the  links  is  very  long  this  substitution 
of  a  sliding-block  for  a  link  may  be  convenient,  the  radius  of  curva- 
ture of  the  slot  being  equal  to  the  length  of  the  link  replaced.  If 
the  link  is  of  infinite  length  the  centre  line  of  the  slot  becomes  a 
straight  line,  and  the  motion  of  the  block  is  then  a  rectilinear 
translation. 

The  mechanisms  shown  in  Figs.  69,  70,  71,  and  72  represent 
this  modified  form  of  the  four-link  chain,  or  what  Reuleaux  has 
called  the  "slider-crank  chain."  This  mechanism  is  so  prominent 
in  practical  machine  construction  that  it  will  be  treated  in  detail. 

106.  Crank  and  Connecting-rod. — The  connection  between  the 
piston  and  crank  of  the  ordinary  direct-acting  engine,  as  shown 
in  Fig.  27,  is  one  of  the  most  important  examples  of  the  modified 
four-link  chain.  In  this  case  the  motion  of  the  piston  and  cross- 
head  relative  to  the  frame  is  equivalent  to  that  of  a  link  of  infinite 
length.  The  piston  and  cross-head,  being  rigidly  connected,  are 
kinematically  one  piece;  though  we  may  be  only  concerned  with 


LINEWOEK.  193 

the  motion  of  the  piston,  it  is  often  convenient  to  speak  of  the  mo- 
tion of  the  cross-head,  which  is  identical  with  that  of  the  piston. 

With  the  usual  arrangement  the  path  of  the  cross-head  is  in  a 
line  which  passes  through  the  crank-centre,  and  this  line  will  be 
spoken  of  as  the  centre  line.  This  is  not  a  necessary  condition, 
and  it  is  sometimes  departed  from  with  a  result  that  will  be  dis- 
cussed in  a  later  article.  Unless  otherwise  stated  it  is  to  be  under- 
stood that  this  ordinary  arrangement  is  meant. 

In  the  steam-engine  the  reciprocating  piston  is  the  driver  and 
the  crank  is  the  follower.  In  case  of  an  air-  compressor  or  power- 
pump,  the  reverse  is  the  case;  but  the  relative  motion  of  the  mem- 
bers is  not  affected  by  this  relation,  as  it  depends  simply  upon  the 
proportions  of  the  mechanism. 

The  crank  usually  has  a  uniform  rotation,  or  approximately 
such,  and  this  condition  will  be  assumed  in  the  following  discus- 
sion. It  is  evident  that  the  cross-head  (piston)  must  come  to  rest 
as  the  crank-pin  passes  the  line  of  centres  (dead-centre  positions). 
Its  motion  is  accelerated  as  it  leaves  either  extreme  position,  attain- 
ing a  maximum  velocity  near  the  middle  of  the  stroke,  followed  by 
retardation  (negative  acceleration)  through  the  latter  part  of 
the  stroke.  The  motion  of  the  reciprocating  parts  is  approxi- 
mately harmonic,  departing  from  true  harmonic  motion  more  as 
the  ratio  of  connecting-rod  length  to  crank  length  becomes  smaller. 

The  position  of  the  crosshead,  c  (Fig.  164),  for  any  crank  posi- 


tion  C,  is  obtained  graphically  by  taking  a  radius  equal  to  the 
length  of  the  connecting-rod  Cc,  and,  with  a  centre  at  the  given 
crank  position,  cutting  the  path  of  the  cross-head  by  an  arc  of  this 


194  KINEMATICS  OF  MACHINERY. 

radius.  In  a  similar  way  the  crank  position  corresponding  to  any 
cross-head  position  is  found  by  taking  a  centre  at  the  given  cross- 
head  position  and  cutting  the  crank-circle  with  an  arc  of  the  above 
radius.  This  last  process  gives  two  intersections,  one  above  and 
one  below  the  line  of  centres,  as  it  should;  for  the  cross-head  passes 
the  same  point  in  its  path  during  the  forward  and  return  strokes, 
both  of  which  are  accomplished  during  a  single  revolution  of  the 
crank.  If  a  series  of  equidistant  cross-head  positions  are  taken,  it 
is  evident  that  the  corresponding  crank-pin  positions  will  not  be 
equidistant,  and  vice  versa.  That  is,  equal  increments  of  cross- 
head  (piston)  motion  do  not  impart  equal  increments  of  motion  to 
the  crank. 

In  drafting-room  practice  these  graphic  methods  of  finding 
simultaneous  crank  and  cross-head  positions  are  usually  most  con- 
venient; but  sometimes  it  is  desirable  to  use  analytical  expressions 
for  the  relations  between  the  crank  and  connecting-rod. 

The  more  important  kinematic  relations  will  be  derived,  trigo- 
metrically,  using  the  following  notation  (see  Fig.  164) : 

Centre  of  crank-circle  =  Q. 

Centre  of  crank-pin  =  C. 

Centre  of  cross-head  pin  =  c. 

Length  of  connecting-rod  =  I. 

Length  of  crank  =  r. 

Ratio  of  connecting-rod  to  crank  (?  —•  r)  =  n. 

Crank  dead-centres  =  A  and  B. 

Corresponding  cross-head  positions  (ends  of  stroke)  =  a  and  b. 

Mid-stroke  cross-head  position  =  q. 

Mid-crank  positions  (quarters)  =  M  and  M*. 

Simultaneous  cross-head  position  =  m. 

Crank  angle,  ahead  of  A,  =  B. 

Corresponding  angle  of  connecting-rod  with  line  of  centres  =  0. 

Drop  a  perpendicular,  Ck,  from  0  upon  the  centre  line;  then 
Ck  =  r  sin  8  =  1  sin  0  ;  QJk  =  r  cos  0  ;  clc  =  V?  -  (CKf 
=  V?  -  r*  sina  6. 


LINKWORK.  195 

For  any  crank  angle,  0,  the  distance  from  c  to  Q  =  ck  -{-  Qk  = 


Vl*  —  r*  sin2  8  +  r  cos  0. 

When  C  is  at  the  quarter  (M  or  Mf),  c  is  at  in,  a  distance  from 
its  mid-position  =  mq  =  Qq  —  Qm  =  I  —  Vl*  —  r3  =  nr  — 
r  Vn*  —  1  =  r  (n  —  V  n*  —  1),  a  quantity  which  increases  as  w  de- 
creases, and  equals  zero  when  n  —  infinity. 

It  is  seen  from  the  above  expression  that  when  the  crank  has 
rotated  through  90°  from  A,  the  cross-head  has  moved  through  more 
than  half  its  stroke;  while  for  the  next  90°  crank  rotation  the 
cross-head  moves  through  less  than  half  its  stroke.  It  follows  that, 
with  uniform  rotation  of  the  crank,  the  half-stroke,  aq,  is  made  in 
less  time  than  the  half-stroke,  qb,  this  variation  decreasing  as  the 
connecting-rod  length  increases.  The  influence  of  this  angularity 
of  the  rod  on  steam  distribution  will  be  seen  to  be  important,  when 
the  subject  of  valve  motions  is  studied. 

In  the  illustrations  of  velocity  diagrams  (see  Art.  41  and  Figs. 
69  and  76)  it  was  shown  that  the  ratio  of  the  linear  velocities  of 
the  crank-pin  and  cross-head  is  equal  to  the  ratio  between  the  length 
•of  the  crank  and  that  segment  of  a  perpendicular  to  the  line  of 
•centres  through  the  shaft  which  lies  between  the  centre  line  and 
-the  line  of  the  connecting-rod,  the  latter  prolonged  if  necessary. 
Thus,  in  Fig.  164,  if  QC  represents  the  velocity  of  the  crank-pin, 
to  some  scale,  s  =  Qt  is  the  velocity  of  the  piston  (to  the  same 
scale)  when  the  crank  is  at  C.  If  the  crank-pin  velocity  can  not 
be  represented  conveniently  to  this  scale,  lay  oif  Qv,  along  the  line 
of  the  crank,  to  represent  its  velocity,  and  draw  vtr  parallel  to  cC; 
then  Qtf  =  sf  is  the  required  velocity  of  the  piston  to  the  scale 
assumed ;  and  the  value  thus  obtained  for  the  piston  velocity  can 
be  used,  as  in  Fig.  76,  for  constructing  the  velocity  diagram. 

Another  method  of  determining  the  ordinates  of  the  velocity 
diagram  is  shown  in  Fig.  165.  With  this  method,  Cv  is  laid  off  on 
the  extension  of  the  line  of  the  crank  to  represent  the  crank-pin 
velocity  to  a  convenient  scale ;  an  ordinate  is  erected  at  c,  and  this 
is  cut  by  drawing  the  line  w'  parallel  to  the  connecting-rod;  then 


1S6 


KINEMATICS  OF  MACHINERY. 


Fig.  165. 


cvf  gives  the  velocity  of  the  cross-head  for  this  phase  (Art.  40).    The 

linear  velocities  of  Cto  c  are  in  the 
ratio  of  OC  to  Oc ;  as  C  and  c 
are  two  points  in  the  connecting- 
rod,  which  at  the  instant  has  a 
motion  equivalent  to  a  rotation 
about  the  instant  centre  0;  hence 
the  velocity  of  C  is  to  that  of 
c  as  OC  is  to  Oc.  The  construction  of  the  complete  diagram  will 
readily  be  seen  from  the  figure.  The  method  is  the  same  as  that 
used  for  the  four-link  chain  in  Fig.  77. 

From  Fig.  164  it  will  be  seen  that  the  velocity  of  the  piston  is 
equal  to  that  of  the  crank-pin  when  s  =  r.  There  are  two  positions 
of  the  piston  in  each  stroke  where  this  condition  is  fulfilled.  The 
first  of  these  positions  is  shown  in  Fig.  166,  where  cC,  produced, 
passes  through  M\  hence  s  =  QM  =  r.  This  equality  of  velocities 
can  be  seen  directly  by  locating  the  instant  centre  0;  for  OCc- 
is  similar  to  QCM  at  this  phase;  hence  OC '=  Oc,  and  the  linear 
velocity  of  c  =  linear  velocity  of  C.  The  same  relation  can  be 
shown  by  resolution  of  the  velocities. 


k     R 


When  C  (Fig.  167)  coincides  with  M,  s  =  QO=  QM,  and  the 
crank-pin  and  the  piston  have  the  same  velocity.  Between  these 
two  positions  of  equal  crank-pin  and  piston  velocity,  the  piston 
moves  faster  than  the  crank-pin ;  for  s  is  greater  than  r.  (See  Fig. 

169.) 

The  second  of  these  positions  of  equal  velocity  is  always 


LINKWORK.  197 

at  which  the  crank  is  perpendicular  to  the  line  of  centres,  and 
is  independent  of  the  ratio  of  connecting-rod  to  crank;  provided 
this  ratio  is  greater  than  unity.  The  first  position  is  a  function 
of  this  ratio;  and  the  crank-angle  corresponding  to  this  phase  is 
found  as  follows  (Fig.  166) :  Let  fall  Qe  perpendicular  to  CM,  then 
as  QMC  is  isosceles,  QMe  and  QCe  are  equal  triangles,  and  the 
angle  eQO  =  eQM  =  0,  .-.  MQC  =  20.  .-.  0  =  90  -  20.  Ck  = 

1  sin  0  =  r  sin  0  =  r  sin  (90—20)  =  r  cos  2  0, .-.  -  sin  0  =  n  sin  0 

=  cos  20;  =1  —  2  sin2  0,  . '.  2  sin9  0  +  n  sm  0  =  1;  dividing  by 

2  and  completing  the  square : 

A}  7?  '  fl 

—  =i  +  jg- 


and  sin  0  =  ±  J-       8  +  »•  —.  j 

The  double  sign  of  the  radical  may  be  dropped,  for  if  the 
minus  sign  be  taken,  with  any  value  of  n  greater  than  1,  we 
would  get  a  value  for  sin  0  numerically  greater  than  1,  which  is 
impossible.  Taking  the  plus  sign: 

sin  0  =  i  i  i/(8  +  n9)  -  n  \ 
As  sin  6  =  n  sin  0, 


sn      = 

This  form  is  convenient  for  graphical  solution  as  follows  (Fig.  168) : 
Lay  off  distance  AB  =  n  to  a  scale  of  r  =•  1,  and  erect  a  perpen- 
dicular BC  =  2.828  to  the  same  scale.  Connect  A  and  C,  then  the 
hypothenuse  AC  =  4/(2.828)3  +  n\  With  A  as  a  centre  and  AB 
(=  n)  as  a  radius,  describe  arc  BD,  cutting  AC  in  D. 


DC=  i/(2.828)' +  raa  -  n. 
DG  X  -  =  sin  6>;  or  DC  X      =  r  sin  0  =  tffc    (Fig.  166.) 


198 


KINEMATICS   OF  MACHINERY. 


Between  the  two  positions  of  the  crank  at  which  the  crank-pin- 
velocity  equals  the  velocity  of  the  reciprocating  parts,  the  velocity  of 
the  latter  is  greater  than  that  of  the  crank,  as  noted  above.  These 


reciprocating  parts  (piston,  piston-rod  and  cross-head)  have  very 
nearly  the  maximum  velocity  at  the  position  where  the  connecting 
rod  and  crank  form  a  right  angle  at  0  (Fig.  169).  The  true  phase 
for  the  maximum  velocity  of  the  piston  is  a  little  later  than  the 
above  position;  but  it  is  difficult  to  locate  this  exact  position,  and 
with  the  proportions  of  crank  and  connecting-rod  used  in  ordinal y 
engines  (I  -~  r  =  n  =  from  4  to  6  usually),  this  error  is  of  no  prac- 
tical account,  and  the  approximation  is  much  more  conveniently 
used.  To  find  the  crank  position  (Fig.  169)  at  which  the  crank  is 
perpendicular  to  connecting-rod,  erect  Ac'  perpendicular  to 
the  line  of  centres  at  A,  equal  to  the  length  of  the  rod,  and  connect 
c'  with  Q.  The  intersection  of  c'Q  with  the  crank-circle  locates 
the  required  position  of  the  crank,  C ;  for  Ac'  =  Cc\  AQ  =  CQ\ 
and  in  the  two  triangles  AQc'  and  CQc,  the  angle  A QC=  0  is 
common.  "When  two  sides  and  the  corresponding  angle  of  two  tri- 
angles are  equal  the  triangles  are  equal;  therefore,  as  QAc'  is  a  right 
angle  by  construction,  cCQ  is  also  a  right  angle,  and  C  is  the  posi- 
tion of  the  crank-pin  required.  The  phase  at  which  the  piston  has 
its  maximum  velocity  is  of  importance  in  certain  problems  relating 
to  the  mechanics  of  the  steam-engine,  for  it  is  the  phase  at  which 
the  acceleration  of  the  reciprocating  parts  is  zero.  In  high-speed 
engines  the  acceleration  of  the  reciprocating  parts  has  a  very  im- 
portant bearing  upon  pressures  transmitted  from  the  piston  to  the 
crank. 


L1NKWORK.  199 

107.  The  Eccentric. — The  eccentric  is  a  modified  crank,  and  all 
that  has  been  said  in  the  preceding  article  applies  to  the  eccentric 
and  rod.     If  the  crank-pin  be  grad- 
ually enlarged,  its  throw  remaining 

unchanged,  the  motion  transmitted 
to  a  given  connecting-rod    is    un- 
altered.    Fig.    170   shows    such    a 
crank-pin     enlarged     in     diameter  \ 
until  it  includes  the    shaft,   and  it  gives    the   familiar   eccentric 
and  rod. 

The  throw  of  the  eccentric  is  the  radius  of  the  equivalent  crank, 
QC\  or  it  equals  the  distance  from  the  centre  of  the  eccentric  to 
the  centre  of  the  shaft  about  which  it  turns. 

The  enlargement  of  the  pin  increases  the  friction,  although  it 
has  no  kinematic  effect.  The  eccentric  is  a  useful  expedient  when 
a  crank  of  small  throw  is  required  which  cannot  be  conveniently 
located  at  the  end  of  the  shaft,  for  under  such  conditions  the  ordi- 
nary connecting-rod  would  "interfere"  with  the  shaft  unless  a 
double-throw  crank  were  used,  and  this  latter  form  would  weaken 
the  shaft  by  cutting  into  it,  besides  being  a  more  expensive  con- 
struction. For  these  reasons  the  eccentric  is  very  commonly  em- 
ployed for  operating  the  valves  of  engines,  imparting  a  reciprocat- 
ing, and  nearly  harmonic,  motion  to  them. 

108,  Connecting-rod  of  Infinite  Length. — It  has  been  seen  that 
the  stroke  of  the  cross-head  (Fig.  164)  equals  the  diameter  of  the 
crank-pin  circle,  =  2r ;  and  that  the  obliquity  of  the  connecting- 
rod  distorts  the  cross-head  motion  from  a  true  harmonic  motion, 
causing  the  half-stroke  farthest  from  the  shaft  (at  the  head  end  of 
the  cylinder)  to  be  made  in  less  time  than  is  taken  by  the  half- 
stroke  nearest  the  shaft  (the  crank  end).     It  was  shown  in  Art.  106 
that  the  displacement   of   the  piston  from  mid-stroke,  when  the 
crank  is  at  either  "quarter/7  or  6  =  90°  (measured,  in  Fig.  164,  by 
qm)  is  less  as  the  connecting-rod  is  made  longer,  relative  to  the 
crank;  or  as  I  ~-  r  =  n  becomes  greater. 

If  the  rod  were  of  infinite  length,  the  cross-head  would  be  at  the 
middle  of  its  stroke  when  the  crank  is  at  the  quarter  (d  =  90°); 


200 


KINEMATICS   OF  MACHINERY. 


r 


for  it  was  shown  that  mq  =  I  —  Vl*  —  r*  ;  hence,  mq  =  I  —  I  =  o, 
when  the  length  of  the  rod  is  infinity.     It  is,  of  course,  impossible 

to  have  a  rod  of  infinite  length ; 
but  it  was  shown  in  Art.  105  that 
the  cross-head  and  guides  give  the 
equivalent  of  an  infinite  length  of 
link  as  to  one  member  of  the  four- 
link  chain;  and  the  slotted  rod 
and  block  of  Fig.  171  may  be  in- 
troduced as  an  equivalent  to  an 
infinite  connecting-rod.  That  is, 
this  mechanism  is  the  equivalent  of  the  four-link  chain  with  two 
links  of  infinite  length. 

With  the  mechanism  of  Fig.  171  the  crank  is  acted  upon  by  the 
slotted  rod  through  the  block.  The  component  of  the  motion  of 
the  crank-pin,  which  is  normal  to  the  acting  faces  of  the  yoke,  equals 
the  motion  of  the  rod.  This  normal  component  is  seen  to  equal 
the  motion  of  the  crank-pin  multiplied  by  cos  0  ;  and  as  0  =  90°  — 
0,  cos  0  =  sin  0,  hence  the  velocity  of  a  piston  attached  to  the 
slotted  rod  is  equal  to  v  sin  0,  when  v  is  the  velocity  of  the  crank. 
The  piston  velocity  is  a  maximum  when  sin  0  is  a  maximum  (assum- 
ing the  crank  to  rotate  uniformly)  ;  or  when  0  =  90°.  At  this 
phase  the  piston  and  crank  have  the  same  velocity,  since  sin  0  =  1. 
This  agrees  with  the  statement  in  Art.  106  that  the  velocity  of  the 
piston  always  equals  that  of  the  crank  when  0  =  90°.  With  the 
finite  rod  there  is  another  crank  position,  for  a  smaller  value  of  0, 
at  which  this  equality  also  exists,  and  between  these  two  crank  posi- 
tions the  piston  velocity  is  greater  than  that  of  the  crank.  As  the 
rod  is  increased  in  length,  these  two  positions  for  equality  approach 
each  other,  the  first  one  more  nearly  corresponding  to  0  =  90°. 
With  the  infinite  rod  the  two  phases  for  equality  coincide,  and  the 
phase  for  maximum  velocity,  which  in  the  general  case  lies  between 
them,  also  falls  at  0  =  90°,  as  seen  above.  These  conditions  will 
be  found  to  harmonize  with  the  general  relations  deduced  above. 
Fig.  171  indicates  the  application  of  this  mechanism,  as  it  is  some- 


LINKWORK. 


201 


times  made  to  steam  fire-engines  and  other  steam-pumps.  P  indi- 
cates the  pump-cylinder  and  S  the  steam-cylinder.  The  crank- 
shaft carries  the  fly-wheel. 

The  practical  effect  of  this  "rod  of  infinite  length,"  or  the 
Scotch  yoke,  as  it  is  frequently  called,  is  to  make  a  more  compact 
mechanism  than  would  be  obtained  with  a  finite  rod  of  ordinary 
length  ;  for  the  jdistance  between  the  "  glands  "  of  the  stuffing- 
boxes  on  the  two  cylinders  needs  be  only  equal  to  the  stroke  plus 
the  outside  width  of  the  slotted  yoke,  with  a  small  allowance  each 
side  for  clearance. 

The  slid  ing-block  is  not  an  essential,  kinematically,  as  the 
crank-pin  could  act  directly  on  the  faces  of  the  slot  ;  but,  as  shown 
in  Art.  28,  it  is  generally  desirable,  when  the  conditions  will  per- 
mit, to  use  surface  contact  instead  of  line  contact,  thus  distribut- 
ing the  pressure  transmitted  over  a  larger  area. 

The  sliding  of  the  block  in  the  slotted  member  produces  friction 
and  resultant  wear,  which  is  not  so  easily  overcome  as  in  a  pin  con- 
nection ;  and  the  ordinary  form  of  connecting-rod  is  therefore  pre- 
ferred as  an  engine  connection  when  the  utmost  compactness  is  not 
a  leading  consideration. 

109.  Connecting-rod  of  Length  Equal  to  Crank. — If  the  connect- 
ing-rod is  of  a  length  equal  to  the  throw  of  the  crank,  as  in  Fig.  172, 
these  two  members  always  form  an 
isosceles  triangle,  with  the  inter- 
cept on  the  centre  line  between  the 
cross-head  and  shaft  as  a  base.  The 
distance  Qa  =  r  +  I  =  2r,  and  6,  the 
end  of  the  stroke  next  to  the  shaft, 
coincides  with  Q.  In  this  arrange-  d 
ment,  c  would  be  drawn  from  a  to 
Q  during  a  crank  movement  AM 
and  the  displacement  from  the 
centre  of  stroke,  due  to  angularity 
of  the  rod,  =  A  Q  =  r.  If  the  cross-head  comes  to  rest  at  Q 
when  C  reaches  M,  with  any  farther  motion  of  the  crank  the  con- 


202  KINEMATICS   OF  MACHINERY. 

necting-rod  would  simply  rotate  around  Q  with  the  crank.  If  the 
cross-head  continues  to  move  in  the  line  aQ,  produced  beyond  ft 
the  crank  movement  MB  would  drive  the  cross-head  to  d,  a  distance 
from  Q  =  2r,  and  the  total  stroke  of  the  cross-head  would  =  4r. 
•The  inertia  of  the  cross-head  as  it  approaches  Q  would  tend  to  pro- 
duce this  effect  ;  but  such  a  motion  can  be  made  positive  by  the 
mechanism  shown  by  the  extension  of  cC  to  c'.  In  this  form  the 
rigid  rod  ccr  (  =  2r)  is  pivoted  at  its  centre  to  the  crank-pin,  and 
cross-heads  at  the  ends  of  the  rod  move  in  guides  at  right  angles  to 
each  other  which  intersect  in  Q.  When  C  is  at  A,  c  is  at  a,  and  c' 
is  at  Q.  As  Amoves  to  M,  c  moves  to  Q, and  c'  moves  to  af  ;  then, 
as  the  motion  of  C  continues  to  the  position  B,  c  passes  Q,  moving 
to  d,  and  c'  returns  to  Q.  As  (7  passes  B  and  moves  to  M',  c'  passes 
from  Q  to  d',  and  c  returns  to  Q.  During  the  completion  of  the 
crank  revolution,  C  moves  from  M'  to  A ,  c  moves  from  Q  to  «,  and 
c'  returns  to  Q,  completing  the  cycle. 

At  any  phase  the  distance  of  c'  from  Q  corresponds  to  Qt,  of 
Fig.  164,  and  hence  is  proportional  to  the  velocity  of  c\  likewise, 
Qc  is  proportional  to  the  velocity  of  c'  at  any  phase.  In  this  mech- 
anism there  is  a  transverse  stress,  as  well  as  tension  or  compression 
on  the  rod  cc'. 

110.  Path  of  Cross-head  Passing  Outside  of  Shaft-centre. — If 
the  line  of  cross-head  motion,  gh  (Fig.  173),  does  not  pass  through 
Q,  the  motion  is  modified  as  follows : 


To  find  the  ends  of  stroke  a  and  b :  first,  take  a  radius  =  I  -f-  yi 
with  a  centre  at  Q,  and  cut  gli  in  a  ;  second  take  a  radius  =  I  —  r, 
with  the  same  centre,  and  cut  gh  in  b ;  the  required  points  are  a 


LINK  WORK.  203 

and  b,  and  the  corresponding  crank  positions  are  QA  and  QB. 
The  stroke  from  a  to  b  is  made  while  the  crank  moves  through  the 
arc  AMB  ;  and  the  return  stroke  takes  place  as  C  moves  through 
the  arc  BM'A.  If  the  crank  motion  is  uniform,  the  forward  and 
return  strokes  are  made  in  unequal  times,  and  this  mechanism 
gives  one  form  of  "quick-return  motion."  If  it  is  required  to  de- 
sign such  a  quick-return  motion,  the  relative  times  of  forward  and 
return  strokes  being  given:  draw  the  crank  circle  and  divide  its 
circumference  into  two  arcs  having  the  required  ratio,  AMP., 
BM'A.  Extend  the  radii  through  these  points  of  division  A  and 
B,  in  directions  QA  and  BQ\  then  lay  off  from  Q  on  the  exten- 
sion of  QA,  I  4-  r,  locating  a ;  and  on  BQ  lay  off  I  —  r  from  Q, 
giving  6 ;  a  and  6  are  the  ends  of  the  stroke,  and  ab  is  the  line  of 
cross-head  motion. 

In  the  Westinghouse  engine  the  above  construction  is  applied  ; 
that  is,  the  line  of  piston  travel  passes  to  one  side  of  the  shaft- 
centre.  Two  cylinders  are  placed  side  by  side,  with  connecting-rods 
acting  on  cranks  which  are  opposite  each  other  (180°  apart).  This 
engine  is  single-acting,  steam  acting  on  each  piston  only  during  its 
downward  stroke  ;  therefore,  by  giving  the  quick  return  to  the  up- 
ward stroke,  one  piston  makes  its  exhaust-stroke  and  takes  steam 
again  before  the  other  piston  has  quite  completed  its  "working" 
stroke  ;  thus,  there  is  no  period  at  which  the  rotative  effort  is  abso- 
lutely zero.  Furthermore,  the  greatest  angularity  of  the  connect- 
ing-rod occurs  on  the  exhaust-stroke,  and  for  a  given  length  of  con- 
necting-rod, the  maximum  obliquity  of  action  is  reduced  for  the 
stroke  during  which  steam-pressure  is  acting  on  the  piston.  Or,  to 
state  the  case  somewhat  differently,  the  length  of  the  rod  can  be  re- 
duced lor  a  given  maximum  obliquity  during  the  period  of  heavy 
pressure,  thus  permitting  a  more  compact  construction. 

111.  Motion  of  a  Point  in  the  Connecting-rod  between  the 
Cross-head  and  Crank. — In  certain  valve-mechanisms,  motion  is 
taken  from  some  point  in  a  connecting-rod  (or  eccentric  rod)  other 
than  either  of  the  pin-centres  previously  considered.  Let  P  (Fig. 
174)  be  such  a  point,  the  motion  of  which  it  is  desired  to  find. 


204  KINEMATICS  OF  MACHINERY. 

Find  the  instant  centre  0  for  any  chosen  phase  of  the  rod  Cc.     All 

points  of  the  rod,  at  the  instant,  rotate 
about  0  with  the  same  angular  velocity, 
and  with  linear  velocities  proportional 
to  their  radii.    Hence,  the  linear  velocity 
of  P  is  to  that  of  C  as  OP  is  to  OC.     The 
direction  of  the  motion  of  P  is  perpen- 
dicular to  OP,  as  indicated  by  Pv3. 
A  similar  method  can  be  used  if  the  point  P  lies  beyond  either 
the  crank  or  cross-head  in  an  extension  of  the  connecting-rod. 

This  problem  can  be  solved  by  the  resolution  and  composition  of 
relative  velocities  also,  but  not  so  readily. 

112.  Inversion  of  Crank  and  Connecting-rod  Chain. — It  was 
shown  in  Art.  39  that  a  kinematic  chain  may  have  the  appearance 
of  entirely  different  mechanisms  when  different  members  of  it  are 
held  stationary.  Thus,  Figs.  69,  70,  71,  and  72  show  the  four 
possible  inversions  of  the  crank  and  connecting-rod  chain.  The 
case  of  Fig.  69  has  been  treated  in  preceding  articles  of  this  chapter. 
Fig.  70  represents  the  condition  when  the  former  crank  is  made 
the  fixed  member;  this  case  is  next  in  practical  importance  to  the 
ordinary  crank  and  connecting-rod  mechanism.  This  form  may 
be  used  to  secure  a  variable  angular  velocity  of  a  continuously 
rotating  follower  from  a  uniformly  rotating  driver.  It  somewhat 
resembles  the  drag-link  in  its  action.  In  conjunction  with  another 
linkage  this  mechanism  is  frequently  used  to  produce  a  slow  advance 
and  a  quick  return  of  the  cutter-bar  of  a  shaping-machine. 

The  condition  shown  in  Fig.  71  is,  as  already  pointed  out,  the 
mechanism  of  the  oscillating  steam-engine.  The  case  of  Fig.  72 
has  comparatively  little  practical  application.  Any  of  these  can  be 
readily  analyzed  by  the  instant  centre  method.  The  form  in  which 
a  is  fixed  (Fig.  70),  will  be  treated  in  some  detail,  on  account  of  its 
extended  practical  use  ;  the  others  will  not  be  taken  up  as  special 
forms. 

Fig.  175  shows  a  crank  a  which  rotates  about  0  and  is  pivoted 
to  a  sliding  block  by  the  pin  P.  This  block  fits  a  slot  in  the  arm 


LINKWORK. 


205 


b,  which  rotates  about  0'.  The  stationary  member  d  supports  the 
fixed  centres  0  and  0'.  The  point  P  rotates  in  the  circle  AB  ; 
hence,  its  motion  at  any  instant  is  perpendicular  to  the  radius  PO 
(the  centre  line  of  the  crank  a).  The  velocity,  which  is  usually 
uniform  but  not  necessarily  so,  is  designated  by  v\.  We  may  con- 
sider the  point  P  to  act  upon  the  centre  line  (pitch  line)  of  the 
slotted  member,  as  the  block  does  not  affect  the  kinematic  problem. 


The  point  in  a  which  lies  at  P  has  the  velocity  Pv^  and  the 
point  in  the  slotted  bar  5,  which  is  also  at  P  for  the  instant,  has  the 
velocity  Pvy  As  these  two  velocities  have  equal  components  along 
the  common  normal  to  the  contact  surfaces,  the  normal  component 
of  Pvl  =  Pv.>.  As  a  point  in  a,  P  is  fixed  at  the  distance  OP  from 
the  centre  0.  As  a  point  in  £,  it  travels  back  and  forth  along  the 
pitch  line  of  the  slot,  its  distance  from  0',  or  its  effective  radius, 
varying  from  O'A  to  0' B  as  the  driver  moves  from  A  to  B. 
During  the  next  half-revolution  of  the  driver  (B  to  A)  the  effective 
radius  of  the  follower  decreases  from  O'B  to  O'A,  thus  completing 
the  cycle. 

Only  the  component  of  the  motion  of  the  driving-point  which  is 
normal  to  O'P  can  impart  rotation  to  the  follower.  The  velocity  of 
this  component  is  represented  by  v2  —  vt  cos  </>  (in  which  expression  <£ 
is  equal  to  the  angle  OPOf) ;  because  Vi  and  v2  are  perpendicular  to 


206  KINEMATICS  OF  MACHINERY. 

OP  and  O'P,  respectively.  If  a  circle  be  drawn  on  00'  as  a  diam- 
eter, the  intercept,  Pn,  of  the  centre  line  of  the  follower  (extended 
through  0'  if  necessary),  which  lies  between  P  and  this  circle 
is  to  v9  as  the  constant  radius  of  the  driver  is  to  vr  Or,  :  vt  :  i\  : : 
PO  :  Pn\  for,  as  OnO'  is  an  angle  subtended  in  a  semicircle, 
On  is  perpendicular  to  O'P,  hence  Pn  =  PO  cos  0.  The  velocity 
of  the  driver  may  be  represented  to  a  scale  which  will  make  it  equal 
to  PO)  when  Pn  becomes  the  velocity  vv  If  this  velocity  scale  is 
not  convenient,  vl  may  be  laid  off  from  P  towards  0,  as  Pk,  and 
a  line  kl  drawn  perpendicular  to  O'P  will  give  PI  =  #a,  to  this 
latter  scale. 

113.  Quick-return  Motions. — If  (Fig.  175)  a  sliding  block,  cy 
travels  in  the  path  ef,  which  passes  through  0',  and  is  connected 
to  a  point  C  in  an  extension  of  the  slotted  follower  by  the  rod  Cc9 
it  will  reach  one  end  of  its  stroke  when  the  driving-point  P  is  at 
E,  and  this  block  c  reaches  the  other  end  of  its  stroke  when  P 
is  at  F.  While  P  is  moving  through  the  arc  FEE,  c  moves  from 
e  tof;  while  P  moves  through  the  arc  EAF,  c  makes  its  return 
stroke  from  f  to  e.  Now  if  the  driver  rotates  uniformly  the  times 
of  these  forward  and  return  strokes  are  in  the  ratio  of  the  arcs 
FEE  to  EAF.  This  is,  in  principle,  the  Whitworth  quick-return 
mechanism,  as  it  is  frequently  applied  to  shapers.  The  slow  stroke 
is  used  for  the  cutting  stroke  of  the  tool,  while  the  return  stroke 
is  made  more  rapidly,  thus  economizing  time  and  increasing  the 
capacity  of  the  machine, 

In  designing  such  a  mechanism  the  circle  in  which  P  rotates 
may  be  drawn  with  0  as  a  centre;  then  divide  its  circumference 
by  E  and  F  into  two  arcs  having  the  ratio  desired  for  the  times  of 
the  forward  and  return  strokes.  Draw  a  line  through  EF,  ex- 
tended to  one  side,  and  the  path  of  c  lies  in  this  line.  Drop  a 
perpendicular  from  0  upon  EF  and  its  foot  will  locate  0',  the 
fixed  centre  for  the  slotted  arm.  Take  C  at  a  distance  from  0', 
which  will  give  the  required  length  of  stroke,  and  choose  a  suitable 
length  for  the  connecting-rod  Cc, 

In  practice  C  is  a  pin  which  can  be  set  at  different  distances 


LINKWORK. 


207 


along  a  radius  to  0',  so  that  the  length  of  stroke  of  c  can  be 
varied  to  suit  the  work.  The  pin  C  might  be  placed  on  the  same 
side  of  0'  as  the  slot ;  but  it  is  usually  more  convenient  to  locate 
it  as  in  Fig.  175. 

The  velocity  of  C  is  to  v.A  as  O'C  is  to  O'P,  since  these  are  the 
Velocities  of  two  points  in  one  piece  which  rotates  about  0' . 
The  motion  of  C  is  perpendicular  to  O'C,  as  shown  by  Cvv  To 
find  its  velocity  lay  off  27,  (found  as  above)  and  draw  the  line 
v^0fv9  cutting  the  perpendicular  to  O'C at  v9,  and  giving  Cvt  as 
the  velocity  sought.  To  find  the  velocity  of  c,  erect  at  c  a  per- 
pendicular to  its  path ;  lay  off  Cv3'  =  Cv3  on  the  extension  of  O'C, 
and  draw  a  line  v/v/  parallel  to  the  rod  Cc ;  v±  is  an  ordinate  of 
the  velocity  diagram  of  c.  The  student  should  complete  this 
diagram  for  both  strokes,  by  the  method  indicated. 

When  the  location  of  the  instant  centre  Oac  can  be  determined, 
the  linear  velocity  of  c  corresponding  to  the  given  linear  velocity 
of  P  may  be  determined  directly  by  the  general  method  outlined 
at  the  end  of  Art.  40. 

The  practical  construction  of  the  Whitworth  quick-return  motion 
is  shown  in  Fig.  176,  in  which  the  letters  correspond  to  those  of 
Fig.  175.     The  pin  P  is  attached 
to  a  gear  which  rotates  about  0, 


Fig.  176. 


Fig.  177. 


the  centre  of  a  large  fixed  stud.  The  centre  0'  is  a  pin  secured 
in  the  fixed  stud,  and  the  slotted  member  rotates  about  this  centre 
0'.  The  pin  C  can  be  clamped  at  different  points  along  its  slot 
to  secure  corresponding  lengths  of  stroke  of  c. 


208  KINEMATICS  OF  MACHINERY. 

Another  quick-return  mechanism,  also  much  used  for  shapers, 
is  indicated  by  Fig.  177.  The  slotted  bar  is  pivoted  to  the  frame  at 
0',  and  is  driven  by  the  crank  pin  P,  which  rotates  about  0  ay  in 
the  preceding  case.  The  slotted  bar  vibrates  between  the  positions 
O'e  and  O'f,  reaching  an  end  of  its  stroke  when  its  centre  line  ip 
tangent  to  the  crank-pin  circle;  or  when  the  crank  is  at  either  E 
or  F.  It  will  be  seen  that  the  driver  passes  over  the  arc  FEE  for 
the  forward  stroke,  and  through  the  arc  EAFior  the  return  stroke. 
The  former  arc  is  greater  than  the  latter;  hence  the  times  of  tlie 
strokes  are  in  the  ratio  of  these  arcs,  if  the  driver  rotates  uni- 
formly. 

The  normal  component  only  (v2)  of  the  crank-pin  velocity  («;,} 
transmits  motion  to  the  follower;  and  va  =  vl  cos  0.  in  which  0  is 
the  angle  OPO'.  If  a  semicircle  be  drawn  on  00'  as  a  diameter, 
cutting  O'P  at  n,  Pn  =  OP  cos  0  ;  hence  vl  :  vy  ::  OP  :  Pn;  or  if 
OP  represents  the  velocity  of  the  crank-pin,  Pn  represents  the 
velocity  of  the  driven  point  of  the  slotted  arm  to  tne  same  scale. 

The  upper  end  of  the  slotted  arm  drives  the  cutter-bar  of  the 
shaper  as  indicated,  through  a  pin,  C,  which  is  between  two  parallel 
projections  attached  to  the  cutter-bar.  The  velocity  of  C  is  i\ , 
perpendicular  to  O'C,  and  va  :  v2  : :  O'C  :  O'P.  To  find  this  veloc- 
ity draw  a  line  through  O'v2  extended  till  it  cuts  Cv3  in  vs.  The- 
motion  of  the  tool,  vt,  is  the  horizontal  component  of  #8.  It  dif- 
fers little  from  va ;  but  can  be  easily  found  by  the  graphical  con- 
struction shown. 

The  fundamental  portion  of  this  mechanism  is  a  modified  form 
of  the  one  used  in  the  Whitworth  motion  ;  the  only  difference  be- 
ing that  0',  in  this  case,  lies  outside  of  the  orank-pin  circle;  while 
in  the  other  case  it  lies  inside  this  circle.  This  difference  in  the- 
proportions  causes  the  slotted  bar  to  vibrate  through  a  definite 
angle  in  one  case  while  it  rotates  continuously  in  the  other  case. 
The  methods  of  connection  with  the  ram  of  the  shaper  are  quite: 
different  in  these  two  cases,  as  is  also  the  means  of  changing  the 
length  of  the  stroke.  In  the  second  form  this  change  is  made 
by  changing  the  length  of  the  driving  crank-arm,  means  being 


L1NKWORK.  209 

provided  for  moving  the  pin  nearer  to,  or  farther  from,  its  cen- 
tre, 0.  The  adjustment  can  usually  be  made  without  stopping 
the  machine. 

With  the  Whitworth  device,  the  relative  time  of  forward  and 
return  strokes  is  not  varied  by  changing  the  length  of  stroke.  With 
the  second  mechanism  the  ratio  between  the  times  of  the  forward 
and  return  strokes  is  greatest  with  long  strokes.  The  angle  through 
which  the  driver  passes  for  the  forward  stroke  is  180°  +  Q,  where  0 
is  the  angle  of  vibration  of  the  slotted  bar;  and  during  the  return 
stroke  the  driver  passes  through  180°  -  6.  The  sine  of  -J  6  =  OP 
-f-  00',  and  as  00'  is  a  constant,  0  varies  with  changes  of  OP. 

To  design  this  machine,  decide  upon  the  ratio  of  the  times  to 
be  occupied  in  the  forward  and  return  strokes  for  some  particular 
length  of  stroke.  Draw  the  crank-circle  for  this  particular  stroke 
(Fig.  177)  and  divide  it  into  the  arcs  FBEaud.  EAF,  having  this 
ratio.  Draw  tangents  to  this  circle  at  B  and  F,  and  their  intersec- 
tion locates  0'. 

The  velocity  diagram  is  readily  constructed  for  both  strokes  by 
finding  the  velocity  =  ?'4  for  various  positions  of  the  ram,  by  the 
method  given.  This  diagram  should  be  drawn  as  an  exercise. 

The  crank  and  connecting-rod  when  arranged  so  that  the  centre 
line  passes  outside  of  the  crank-circle  centre  (as  discussed  in  Art. 
110),  may  be  used  for  a  quick  return.  Elliptical  gears  (see  Art. 
46)  are  also  used  for  quick-return  mechanisms. 

114.  Bell-cranks. — Fig.  178  shows  the  method  of  designing  a  bent 
lever,  or  bell-crank,  to  transmit  motion  from  line  OA  to  line  OB,  with 
linear  velocity  ratio  =•  m  -f-  n.  Lay 
off  Oa  =  m  on  0  B,  and  Ob  =  n  on 
OA  ;  complete  the  parallelogram 
Obqa  by  drawing  aa  and  bb  parallel  to 
OA  and  OB,,  respectively,  and  inter- 
secting at  q.  Through  0  and  q  draw 
a  line.  Any  centre,  as  Q,  on  this  line 
may  be  taken  as  the  bell-crank  centre.  From  Q,  drop  perpen- 
diculars QP  and  Qp  on  OA  and  OB',  these  are  centre  lines  of  the 


210 


KINEMATICS  OF  MACHINERY. 


arms  of  the  bell-crank.     The  angular  motion  of  the  arms  should  be 
the  same  on  each  side  of  QP  and  Qp. 

It  will  be  seen  that  any  motion,  either  side  of  the  central  posi- 
tion, will  shorten  the  effective  arms.  To  reduce  the  deviation  of  the 
connected  points  to  a  minimum,  the  lengths  R  and  r  should  be 
greater  than  QP  and  Qp,  respectively,  by  one-half  the  versed  sine 
of  the  angle  described  each  side  of  the  central  position  multiplied 
by  the  respective  radii. 

115.  The  Beam.  —  Fig.  179  indicates  the  beam  of  an  engine  ; 
CO  is  the  line  of  piston  motion;  QF=  distance  of  beam  centre 

from  this  line,  =  d  ;  AB  —  stroke  ; 
A  E  =  J-  stroke  =  s  ;  DF  should  = 
EF,  to  minimize  the  angularity  of  the 
connecting-rod.  The  beam  length  to 
fulfil  this  requirement  is  found  thus: 
QD  =  QA  =  d  ~ 


EF\  but  QAt  = 


, d--^*— ->-' 


Fig.  179. 


~AE~\ 


EF?-\-s\     .'.  d*  +  2d  .EF  +  EF*  =d*  -  2d  .  EF  +  ~EF*  -f  s\ 
.-.    4d  .  EF=s\     .-.  EF  =  DF  =  *"  -r-  4=d  ;  .-.  the  length  of  the 

»• 
beam  =  d  +  DF  =  d  +  TJ- 

116.  Rapid   Change  in  Angular  Velocity  of  the  Follower.  —  A 

means  of  imparting  rapidly  changing  angular  velocity  to  a  follower 

by  the  use  of  the  linkwork  is  shown 

by  Fig.  180.     As  C  reaches  the  vari- 

ous positions  in  its  path  marked  0,  1, 

2,  3,  etc.,  c  lies  at  0',  1',  2',  3',  etc., 

respectively,  in  its  path.     The  prin- 

ciple of  this  arrangement  is  of  fre- 

quent use.     It  is  applied,  as  indicated 

in  the  figure,  in  the  "  wrist-plate  "  of  the  Corliss  valve-gear,  to  give 

a  rapid  opening  of  the  valves  (arid  quick  closing  of  the  exhaust- 

valves)  with  a  small  motion  when  they  are  closed.     In  this  applica- 

tion the  rotating  (vibrating)  valve  is  attached  to  the  arm  0'  c,  and 


$r 


LINKWORK.  211 

the  mechanism  is  so  proportioned  that  the  required  port  opening  is 
given  quickly  to  a  valve  at  one  end  of  the  cylinder,  while  the  valve- 
arm  at  the  other  end  moves  but  little  during  this  period. 

In  general,  the  motion  of  the  follower  c  is  small  compared  to  a 
given  motion  of  the  driver  G  as  the  angle  0'  c  C  approaches  a  right 
angle  and  the  angle  OCc  approaches  0  or  180°.  On  the  other 
hand,  the  relative  motion  of  c  to  C  is  great  as  the  angle  0'  c  C  ap- 
proaches 0  or  180°  and  the  angle  OCc  approaches  90°. 

117.  Straight-line  Motion. — A  large  number  of  linkages  have 
been  devised  to  make  a  point  move  in  a  straight  line  independently 
of  any  planed  guides. 

The  term  Parallel  Motions  is  usually  given  to  such  mechanisms, 
but  straight-line  motions  is  a  more  appropriate  term.  Fig.  181 
shows  what  is  known  as  Watt's  par- 
allel motion.  R  and  r  are  arms  cen- 
tred  at  Q  and  q  ;  Aa  is  a  link  con- 
necting the  free  ends  of  R  and  r,  and 
P  is  a  point  in  Aa  which  traces  an 
approximately  straight  line,  within 
certain  limits  of  the  motion. 

If  R  moves  from  its  central  posi- 
tion, A  is  drawn  to  the  right,  while 

the  accompanying  motion  of  r  carries  a  to  the  left.  The  path  of 
P  is  a  function  of  both  of  these  motions  and  the  result  is  that  P,  if 
properly  located  in  Aa,  moves  very  nearly  in  a  straight  line,  pro- 
vided the  angular  motion  of  R  and  r  does  not  exceed  about  20°. 
The  complete  path  of  P  is  the  "  figure  8  "  shaped  curve  Pmn. 

If  R  =  r,  AP  =  aP.  In  general,  the  segments  AP  and  aP 
are  inversely  as  the  length  of  the  adjacent  arms. 

Watt  used  this  mechanism  to  guide  the  piston-rod  in  place  of 
the  slides  now  generally  employed  ;  but  the  principal  application 
of  "  parallel  motions "  at  present  is  on  steam-engine  indicator 
pencil  motions.  The  Richards  indicator,  the  earliest  of  the  modern 
type,  has  the  Watt  mechanism. 

The  Tabor  indicator  has  a  motion  in  which'  a  curved  guide  is 


212 


KINEMATICS  OF  MACHINERY. 


m 


used  ;  it  is,  therefore,  of  a  different  type  from  the  pure  linkwork 

mechanisms  usually  classed  as  par- 
allel motions.  Fig.  182  indicates  this 
pencil  movement.  It  is  desired  that 
the  pencil  point  P  shall  move  in  a 
right  line,  mm.  It  is  evident  that 
the  curved  guide  nn  can  be  given 
such  a  form  that  this  will  occur,  and  this  curve  can  be  found  by 
moving  P  along  mm,  tracing  the  curve  nn  by  the  point  p.  Hav- 
ing found  nn,  a  circular  arc  may  be  found  which  agrees  closely 
with  it,  within  the  range  of  motion  ;  and  if  the  centre  of  this  arc 
be  at  d,  a  link  dp  can  be  substituted  for  the  curved  guide  nn. 
An  arrangement  similar  to  this  substitution  is  used  on  the  Thomp- 
son indicator.  If  a  moved  in  a  straight  line,  instead  of  in  the  arc, 
yy  ;  if  p  were  at  the  centre  of  Pa  ;  and  if  dp  =  Pp  =  pa,  the 
mechanism  would  be  the  same  as  that  shown  in  Fig.  172,  and  the 
result  would  be  an  exact  straight-line  motion  ;  requiring  a  straight 
guide,  however,  for  the  point  a. 

The  Crosby  indicator  has  a  pencil  mechanism  similar  to  that  of 
Fig.  183.  If  P  be  moved  in  a  straight  line  mm,  p  (a  point  in  the 
link  be)  traces  a  curve  ;  the  bridle  link  dp  is  one  that  will  give  an 
arc  most  nearly  approaching  this  curve. 


Fig    184. 


Peaucellier's  straight-line  motion  is  exact,  and  it  is  a  pure  link- 
work.  It  is  shown  in  Fig.  184.  Two  equal  links  a  and  b  have  a 
fixed  centre,  Q.  The  links  d,  e,  f,  g  are  equal ;  and  c,  with  a  fixed 
centre  at  q,  equals  the  distance  Qq.  P  is  constrained  to  move  in 
the  straight  line  mm. 


LINKWORK. 


213 


118.  Pantographs. — There  are  various  linkages  in  which  if  one 
point  ia  made  to  travel  in  any  path,  some  other  point  will  be  con- 
strained to  describe  a  similar  path,  enlarged  or  reduced. 

Fig.  185  shows  one  such  arrangement  in  which  Pa  =  bQ;  and 
aQ  =  Pb.  These  links  form  a  parallelogram  which  has  a  fixed 
centre  at  Q.  A  bar  cd  is  attached  to 
aQ  and  Pb  parallel  to  Pa,  and  the 
point  p,  in  cd,  which  lies  on  the  line 
•connecting  P  and  Q,  will  move  in  a 
path  similar  to  that  traced  by  P. 
Suppose  P  to  move  to  P',  then  p 
moves  to  p',  and  from  similar  tri- 
angles, QP:Qp::Qa:  Qc;  also  QP' 
-and  Qc  =  Qc' 


Q   ^ 


Fig.  185 


Qp'::Qa':Qc'',l>utQa=Qa', 

QP  :  Qp  ::  QP'  :  Qp',  hence  the  distance  of  p 
from  Q  is  proportional  to  the  distance  of  P  from  Q.  As  p  always 
lies  in  the  line  QP  (because  QaP  and  Qcp  are  similar  triangles), 
the  angular  motion  of  p  about  Q  is  equal  to  the  angular  motion  of 
P  about  Q.  Any  path  of  P  is  determined  by  its  angular  motion 
about  Q  and  its  radius  vector  to  Q  as  a  pole;  as  the  angular  motion 
of  P  and  of  p  about  Q  are  seen  to  be  equal  for  any  motion  of  either 
of  these  points,  and  as  the  radius  vector  of  p  bears  a  constant  ratio 
to  that  of  Pj  the  path  of  p  is  similar  to  that  of  P. 

A  form  of  pantograph,  called  the   "lazy-tongs,"  is  shown  in 
Fig.  186.     It  is  frequently  used  to  reduce  the  piston  motion  of  an 


Fig.  186.  Fig.  187. 

•engine,  in  using  the  indicator.  P  is  attached  to  the  cross-head,  and 
the  indicator  cord  is  attached  at  p.  The  practical  objection  to  this 
contrivance  is  the  great  number  of  joints,  and  consequent  liability 
to  lost  motion  from  wear. 


214 


KINEMATICS  OF  MACHINERY. 


Fig.  187  shows  another  pantograph  for  the  same  use.  P  is  at- 
tached to  the  cross-head,  and  the  cord  is  attached  at  p  as  before. 
With  either  of  these  arrangements  the  point  p  must  lie  in  the  line 
connecting  P  and  Q,  and  the  cord  must  be  led  off  parallel  to  the 
cross-head  motion. 

Watt  combined  the  pantograph  with  his  straight-line  motion  so* 
that  the  piston-rod,  air-pump  rod,  and  feed-pump  rod  were  all 
guided  in  straight  lines  by  means  of  one  combination  of  links. 

119.  Hooke's  Coupling,  or  the  universal  joint,  is  used  for  con- 
necting two  shafts  which  intersect.  It  is  equivalent  to  what  Reu- 
leaux  calls  the  four-link  conic  chain — that  is,  to  a  four-link  chain, 
in  which  the  pivots  are  not  parallel  as  in  the  ordinary  case  already 
treated,  but  their  axes  lie  in  radii  of  a  sphere.  Every  point  moves 
in  the  surface  of  a  sphere,  instead  of  in  a  plane.  In  its  typical  form 
(Fig.  188),  each  shaft  has  a  forked  end,  and  the  two  forks  are 
united  by  an  equal  armed  cross  aft,  cd,  or  its  equivalent.  The 


Fig.  188. 


Fig    189. 


shafts  Mm  and  Nn  and  the  arms  of  the  cross  (the  axes  of  the  pivots) 
intersect  in  a  common  point  0.  If  only  one  half  of  each  fork  be 
considered,  as  mb  of  Mm  and  nd  of  Nn,  and  these  are  assumed  to 
be  connected  by  the  spherical  link  bd  equal  to  the  fixed  -distance 
between  the  two  adjacent  points  of  the  cross,  a  four-Jink  conic 
chain  is  produced  in  which  the  axes  of  all  the  turning  pairs  inter- 
sect in  0.  With  this  arrangement  the  fork  could  be  omitted,  and 


LINK  WORK.  215 

we  would  have  the  kinematic  equivalent  of  the  original  mechan- 
ism. 

The  driver  Mm  and  the  follower  Nn  make  complete  revolutions 
in  the  same  time;  but  the  velocity  ratio  is  not  constant  throughout 
the  revolution. 

If  a  plane  of  projection  be  taken  perpendicular  to  the  axis  of 
Mm,  the  path  of  a  and  b  will  be  the  circle  ABCD  in  Fig.  189.  If 
the  angle  between  th'e  shafts  is  ft,  the  path  of  c  and  d  will  be  a 
circle  which  is  projected  on  the  ellipse  AB'CD',  in  which  OB'  = 
OD'  =  OB  cos  ft  =  OA  cos  ft. 

If  one  of  the  arms  of  the  driver  is  at  A  an  arm  of  the  follower 
will  be  at  B' ';  and  if  the  driver-arm  moves  through  the  angle  6  to  P 
the  following  arm  will  move  to  Q-,  OQ  will  be  perpendicular  to  OP* 
hence  B'OQ  =  B.  But  B'OQ  is  the  projection  of  the  real  angle 
described  by  the  follower.  Qn  is  the  real  component  of  the  follow- 
er's motion  in  the  direction  parallel  to  AC,  which  line  is  the  inter- 
section of  the  planes  of  the  driver's  and  follower's  paths.  The  true 
angle  0,  described  by  the  follower,  while  the  driver  describes  the 
angle  6,  can  be  found  thus:  draw  QR  parallel  to  OB  so  that  Rm  = 
Qn,  then  OR  equals  the  radius  of  the  follower,  and  BOR  =  0  = 
the  true  angle  in  plane  AB' CD'  which  is  projected  as  B' OQ  =  0. 

Now  tan  0  =  Rm  -f-  Om,  and  tan  6  =  Qn  ~-  On ;  but  Qn  =  Rm, 

tan  6  _0m  _  OB  _        1 
•"'  tan0  "~  ~0n  ~  ~OB'  ~~  cos~fi' 

A  tan  0  =  cos  /3  tan  B (1) 

The  angular  velocity  ratio  of  follower  to  driver  is  therefore  found 
as  follows  by  differentiation  of  Eq.  (1),  remembering  that  ft  is  a 
constant  in  this  equation: 

a'  __  d(f)  __  cos  ft  sec*  B  _  cos  ft  sec*  B 

a  ~~  ~dB  =        aec1  0       ~  1  +  tan*  0>   '    '    '     V2) 


216  KINEMATICS  OF  MACHINERY. 

Eliminating  0  by  means  of  (1) 

cos  ft 

a'  cos  0  sec2  0  cos9  9 


a        1  +  cos'  /ftan2  0       cos2  6  -f  sin2  6  cosa  ft 

COS20 

cos  /?  cos  ft 


(3) 


cos2  0  +  sin2  0  (1  -  sina  ft)       1  -  sin2  6  sin*  ft' 
By  a  similar  process  0  could  be  eliminated,  giving 

of_   _  1  —  cos'  0  sin2/? 
a  cos  ft 

It  is  seen  from  (3)  that  #'-*-  or  is  a  minimum  when  sin  0=0, 
or  when  0  =  0,  TT,  etc.,  which  corresponds  to  a  value  of  0  =  0,  TT, 
etc.  The  same  thing  is  seen  from  (4),  which  gives  a  minimum 
value  of  a'  ~  a  when  cos  0  =  1,  or  0  =  0,  TT,  etc.  Also,  a'  -—  a  is 
a  maximum  when  sin  0—1,  or  cos  0=0,  corresponding  to  0  =  90°, 
JTT,  etc.  ;  0  =  90°,  $7r,  etc. 

To  summarize  the  foregoing,  the  follower  has  a  minimum  angu- 
lar velocity,  if  the  driver  has  a  uniform  velocity,  when  the  driving- 
arm  is  at  A  or  C  and  the  following  arm  is  at  Bf  or  D'.  The  fol- 
lower has  a  maximum  angular  velocity  when  the  driving-arm  is  at 
B  or  D  and  the  following  arm  is  at  A  or  C. 

By  using  a  double  joint  the  variation  of  angular  velocity  be- 
tween driver  and  follower  can  be  entirely  avoided.  To  do  this  an 
intermediate  shaft  is  placed  between  the  two  main  shafts,  making 
the  same  angle,  ft,  with  each.  The  two  forks  of  this  intermediate 
shaft  must  be  parallel.  If  the  first  shaft  rotates  uniformly,  the 
angular  velocity  of  the  intermediate  shaft  will  vary  according  to  the 
law  deduced  above.  This  variation  is  exactly  the  same  as  if  the  last 
shaft  rotated  uniformly,  driving  the  intermediate  shaft  ;  therefore, 
as  uniform  motion  of  either  the  first  or  the  last  shaft  imparts  the 
same  variable  motion  to  the  intermediate  shaft,  uniform  motion  of 
either  of  these  shafts  will  impart  (through  the  intermediate  shaft) 
uniform  motion  to  the  other.  This  is  the  combination  used  in  the 
feed-rod  of  the  Brown  &  Sharpe  milling  machines  and  elsewhere. 


LIXKWORK.  217 

120.  Ratchets. — The  ratchet-wheel  and  pawl  (Fig.  190)  resemble 
both  the  direct-contact  motions  and  linkwork.  The  driving-pawl 
CP  acts  by  direct  contact ;  but  dur-  ,IL.  m> 

ing  driving  the  action  is  similar  to 
that  of  a  four-link  chain,  consisting 
of  QC,  qP,  PC,  and  the  fixed  link 
Qq.  Such  mechanisms  are  sometimes 
termed  intermittdfti  linkwork 

The  two  centres  Q  and  q  may  co- 
incide, the  pawl-lever  vibrating  about 
the  axis  of  the  wheel.  In  this  case 
there  is  no  relative  motion  between 
the  members  during  the  forward  (working)  stroke. 

The  supplementary  pawl,c/?,  has  a  fixed  centre,  c,  and  its  object  is 
to  prevent  the  backward  motion  of  the  wheel  when  CP  is  not  driving. 

If  pn  is  the  common  normal  to  the  end  of  cp  and  the  tooth  with 
which  it  engages,  there  is  no  danger  of  the  pawl  becoming  dis- 
engaged under  the  reaction  of  the  tooth  upon  it ;  for  the  centres  c 
and  q  are  on  the  opposite  sides  of  the  line  of  action,  and  the  ten- 
dency is  for  the  wheel  to  run  backward  (right-handed  rotation)  and 
for  the  pawl  to  turn  with  a  left-handed  rotation,  which  only  forces 
them  together.  If  the  direction  of  the  common  normal  is  pn' ,  the 
•centres  both  lie  on  the  same  side  of  the  line  of  action,  when  the 
tendency  is  for  both  pawl  and  wheel  to  rotate  in  the  right-handed 
direction,  and  the  pawl  would  be  forced  out  of  contact,  unless  held 
by  friction.  In  a  similar  way  the  normal  Pm  of  the  driving-pawl 
CP  and  the  tooth  on  which  it  acts  should  pass  between  C  and  Q- 

The  pawl  cp  only  prevents  backward  motion  of  the  wheel  after 
Ihe  wheel  has  moved  back  far  enough  to  come  in  contact  with  the 
pawl.  The  amount  of  backward  motion  possible  may  vary  from 
zero  to  the  pitch  of  the  teeth.  This  action  could  be  limited  by 
making  the  teeth  small ;  but  this  would  weaken  the  teeth,  and  the 
•expedient  is  sometimes  adopted  of  placing  several  pawls  side  by  side 
on  the  pin,  c,  the  pawls  being  of  different  lengths.  With  this 
arrangement  the  maximum  backward  motion  may  be  reduced  to  the 
pitch  divided  by  the  number  of  pawls  provided. 


218 


KINEMATICS  OF  MACHINERY. 


Sometimes,  for  feed-motions,  etc.,  the  pawl  and  wheel  are  made 
as  shown  in  Fig.  191.  This  pawl  can  be  reversed  for  driving  in  the 
opposite  direction. 

Fig.  192  shows  a  double-acting  ratchet  by  which  both  strokes  of 
the  lever  drive  the  wheel.  The  locking-pawl  may  be  omitted  in 
this  case. 


Fig. 192 


Fig.  191. 


Frictional  pawls  (Fig.  193)  are  sometimes  used,  in  which  case 
the  wheel  is  made  without  teeth.  The  pawl  grips  the  wheel  by  fric- 
tion during  one  stroke  and  releases  it  on  the  return  stroke.  These 
have  the  advantage  of  being  noiseless,  and  the  angular  motion  of 
the  wheel  for  each  stroke  is  not  restricted  to  some  multiple  of  the 


Fig.  193.  Fig.  194. 

arc  between  two  teeth,  but  the  driving  is  not  positive.  Another 
frictional  pawl  with  a  fixed  centre  at  c  can  be  used  to  prevent 
"overhauling"  of  the  wheel.  The  letters  of  Fig.  193  correspond 
with  those  of  Fig.  190. 


LINKWORK. 


219 


In  the  form  of  frictional  ratchet  shown  by  Fig.  194,  the  wheel  is 
surrounded  by  a  ring,  which  can  be  vibrated  about  the  axis  of  the 
wheel.  One  of  these  members  (either)  has  teeth  of  the  form 
shown;  and  in  the  depressions  formed  by  the  teeth,  rolls,  or  balls, 
are  placed.  Motion  of  the  driver  in  one  direction  causes  these 
rolls  to  bind  the  follower,  while  they  release  it  on  the  return. 
Positive  "  silent"  ratchets  have  been  made  with  a  special  device  for 
holding  the  pawl  clear  of  the  teeth  on  the  return  stroke. 

The  forms  of  ratchets  shown  by  Figs.  190  to  195,  and  numerous 
modifications  of  them,  are  suitable  for  many  cases  requiring  the 
conversion  of  a  reciprocating  action  into  an  intermittent  rotation. 
They  are  especially  convenient  in  feed-mechanisms  when  the  vibra-, 
tions  of  the  driver  are  not  too  rapid.  At  high  speeds  the  shock 
between  the  pawl  and  tooth,  as  the  driving  begins,  may  be  objec- 
tionable, and  the  inertia  of  the  wheel  is  liable  to  make  it  move 
farther  than  desired,  or  to  cause  "  overtravel."  This  last  tendency 
prevents  the  employment  of  the  ordinary  ratchet  when,  as  in  revo- 
lution registers  or  continuous  counters,  a  definite  motion  of  the 
follower  must  be  insured.  A  device  for  such  purposes  is  shown  by 


Fig.  195. 

Fig.  195.  The  lever  to  which  the  pawl  is  attached  has  a  tooth  or 
beak  so  formed  and  placed  that  overtravel  is  impossible.  When 
the  pawl  first  acts  on  a  pin,  another  pin  passes  close  to  the  point  of 
this  beak;  the  beak  then  follows  in  behind  this  pin,  crossing  the 
path  of  pin-motions,  and  thus  limiting  the  motion  of  the  next  pin. 
The  outline  ab  should  be  a  circular  arc  with  Q  as  a  centre,  so  that 


220  KINEMATICS  OF  MACHINERY. 

the  pin  which  it  stops  will  rest  against  it  during  the  return  stroke 
of  the  driver.  Another  device  much  used  for  counters  is  shown 
by  Fig.  196.  The  "star"  wheel  is  driven  through  half  of  its  pitch 
arc  by  the  action  of  the  projection  b  upon  the  tooth  a  during  one 
stroke  of  the  driver,  and  V  acts  upon  the  opposite  tooth  a'  during 
the  return  stroke,  thus  moving  the  wheel  an  equal  distance  in  the 
same  direction.  It  will  be  seen  that  the  motion  of  the  wheel  for  a 
•double  stroke  of  the  driver  is  equal  to  the  angle  between  two  teeth, 
and  if  the  wheel  has  ten  teeth,  it  will  make  a  complete  rotation  for 
ten  double  strokes  of  the  driver. 

121.  Escapements. — The  mechanism  of  Fig.  196  resembles  the 
escapements  used  to  control  the  motion  of  a  train  of  clockwork, 
and  it  might,  with  slight  modification,  be  used  for  such  a  purpose. 
If  the  wheel  A  is  acted  upon  by  a  spring  or  weight  which  tends  to 
rotate  it  continuously  in  the  left-hand  direction,  this  wheel  would 
tend  to  produce  reciprocation  of  the  piece  B.  If  B  is  a  pendulum, 
it  has  a  normal  period  of  vibration  corresponding  to  its  length,  and 
if  the  pendulum  is  so  heavy  that  the  rota- 
tive effort  of  A  cannot  alter  this  period, 
the  pendulum  in  swinging  will  control 
the  motion  of  the  wheel.  The  tendency 
of  the  wheel  to  produce  vibration  of 
the  pendulum  may  be  made  sufficient  to 
overcome  the  frictional  resistance  which  acts 
to  stop  the  pendulum,  and  thus  the  ampli- 
tude of  the  vibrations  is  maintained.  Other 
outlines  of  teeth  for  the  wheel  and  pendu- 
lum are  better,  practically,  and  one  common  form  is  shown  in 
Fig.  197.  The  teeth  of  the  piece  which  vibrates  with  the  pendu- 
lum are  called  pallets. 

Many  modifications  of  the  escapement  have  been  devised  to 
meet  special  requirements.  In  watches  and  other  portable  time- 
pieces a  balance-wheel  is  used  instead  of  the  pendulum  to  regu- 
late the  period  of  the  vibrating  member,  but  all  are  similar  in  their 
general  action. 


CHAPTER  VII. 


WRAPPING-CONNECTORS.     BELTS,    ROPES,  AND   CHAINS. 

122.  Belts,  Hopes,  Chains,  etc. — Flexible  members  are  frequently 
used  for  transmission  of  motion  between  two  pieces  provided  with 
properly  formed  surfaces  upon  which  the  connector  wraps  or  un- 
wraps as  the  action  takes  place.  The  connector  may  be  a  flat  belt 
or  band,  a  rope,  or  a  chain  composed  of  jointed  members  each  one 
of  which  :s  itself  rigid. 

The  great  majority  of  the  practical  applications  in  which  bands 
are  used  for  transmitting  motion  are  those  in  which  the  velocity 
ratio  is  constant.  Figs.  198  and  199  show  pairs  of  wheels  of 


T  M 


Fig.  198.  T 


circular  section  connected  by  bands.  These  evidently  fulfil  the 
condition  of  constant  velocity  ratio,  for  the  segments  (QF and  qF) 
into  which  the  line  of  the  band  cuts  the  line  of  centres  (or  its  ex- 
tension) are  constant;  also,  the  perpendiculars  (R  and  r)  let  fall 
from  the  fixed  centres  upon  the  line  of  the  band  are  constant  (see 
Art.  31).  In  case  exact  motion  through  only  part  of  a  revolution 
(or  at  most  through  a  limited  number  of  revolutions)  is  to  be  trans- 
mitted, the  ends  of  the  bands  may  be  fastened  to  the  wheels.  The 
action,  with  this  condition,  is  positive,  provided  the  direction  of  the 

221 


222  KINEMATICS  OF  MACHINERY. 

motion  is  such  that  the  band  is  always  kept  in  tension.  Thus  in 
Figs.  55  and  56,  the  piece  which  rotates  about  0  must  be  the  driver, 
while  the  one  rotating  about  0'  is  the  follower,  for  transmission  of 
motion  in  the  directions  indicated.  The  motions  of  two  pieces  con- 
nected in  this  way  are  necessarily  of  a  reciprocating  character,  for 
when  the  band  is  all  unwrapped  from  the  follower  the  mechanism 
comes  to  rest,  and  any  farther  motion  most  be  in  the  reverse  direc- 
tion. Such  motion  can  only  be  secured  when  the  former  follower 
becomes  the  driver.  An  example  similar  to  this  case  is  seen  in  a 
hoisting-drum  which  pulls  a  car  up  an  incline.  While  hoisting,  the 
drum  is  the  driver  relative  to  the  car;  but  in  lowering,  the  action 
of  gravity  on  the  car  causes  it  to  turn  the  drum  backward. 

In  most  common  applications  of  flexible  connectors  the  ends  of 
the  band  are  joined  together  and  not  fastened  to  the  wheels,  and 
the  motion  is  continuous;  this  is  commonly  called  an  endless  band. 

In  these  cases  the  motion  is  not  positive,  as  the  bands  may  slip 
(except  when  chains  are  used),  but  usually  very  exact  motion  is  not 
essential  where  these  devices  are  employed. 

It  follows  from  the  demonstration  of  Art.  31,  referred  to  above, 
that  if  wheels  of  circular  transverse  sections  are  connected  by  a  flex- 
ible band  their  angular  velocities  are  inversely  as  their  radii. 

The  effective  radii  are  greater  than  the  radii  of  the  wheels  by 
about  one  half  the  thickness  of  the  band;  but  generally  the  correc- 
tion for  thickness  of  the  band  need  not  be  made  with  thin  flat 
belts. 

The  exact  effective  diameter  is  the  length  of  band  that  will  just 
encircle  the  wheel  divided  by  TT.  When  a  round  cord  or  rope  or  a 
chain  is  used  this  affords  a  convenient  way  to  get  the  effective  or 
pitch  diameter.  Wheels  for  such  ropes  or  cords  have  grooves  cut  in 
the  rims  to  keep  the  band  on  the  "  sheave."  For  hemp  or  cotton 
rope  transmissions  the  grooves  are  given  such  a  form  that  the  rope 
is  wedged  into  them  slightly,  thus  increasing  the  tractive  force. 
With  wire  rope  this  wedging  is  inadmissible,  as  it  would  injure  the 
rope,  and  the  bottom  of  the  groove  has  a  somewhat  larger  radius 
than  that  of  the  rope.  The  bottom  of  the  groove  in  wire- rope  sheaves 


WRAPPING-  CONNECTORS. 


223 


is  usually  lined  with  rubber,  leather,  wood,  or  some  such  material, 
to  increase  the  adhesion  and  save  wear  of  the  cable. 

Fig.  200  shows  the  section  of  the  rim  of  a  sheave  as  commonly 
designed  for  hemp  or  other  fibrous  ropes.  Fig.  201  shows  a  section 
of  rim  employed  with  wire  ropes.  If  sup- 
porting sheaves  or  tighteners  are  required 
in  a  hemp-rope  transmission  the  groove 
is  made  similar  to  that  shown  for  wire 
rope  but  without  the  soft  lining;  for  as 
these  sheaves  are  not  intended  to  transmit  '9'  200-  Fig.  201. 

power,  the  increased  adhesion  due  to  the  wedging  of  the  rope  is  not 
required,  and  unnecessary  wear  of  the  rope  is  avoided  by  making  the 
groove  larger. 

With  chain-bands  the  wheels,  called  "  sprocket "  wheels,  have 
projections  fitting  the  links  of  the  chain  (more  or  less  closely)  to 
prevent  slipping.  With  flat  belts  the  pulleys  have  flat  or  nearly 
flat  faces.  The  forms  of  sprocket-wheels  and  of  the  faces  of  pulleys 
for  flat  belts  will  be  treated  in  later  articles. 

123.  Shifting  Belts.  — If  a  pressure  is  brought  to  bear  upon  the 
advancing  side  of  a  belt  (Fig.  202)  the  belt  is  deflected  in  the  direc- 
tion of  this  force.     As  the  belt  passes  upon  the  pulley, 
each  successive  portion  of  it  passes  upon  a  part  of  the 
pulley  farther  from  the  side  from  which  the  shifting 
J  force  acts,  and  the  belt  takes  up  a  new  position,  as 
shown  by  •  the  dotted  lines.     A  pressure   upon   the 
receding  side  of  the  belt  does  not  have  this  effect, 
-[— |— i    unless   the  force  is  great  enough   to  overcome  the 


T 


|    * —  adhesion  of  the  belt  and  pull  it  over  bodily.     It  must 
be  remembered,  however,  that  the  receding  side  of 
^        the  belt  relative  to  one  pulley  is  the  advancing  side 
Fig.  202.      relative  to  the  other  pulley. 

124.  Crowning  Pulleys.— If  a  flat  belt  is  placed  upon  a  cone 
(Fig.  203)  the  edge  nearest  the  base  of  the  cone  is  stretched  more 
than  the  other  parts,  and  the  belt  tends  to  take  the  position  shown 
by  the  dotted  line.  The  effect  of  this  is  to  shift  the  belt  towards 


224 


KINEMATICS  OF  MACHINERY. 


Fig.  203. 


the  base  of  the  cone,  as  the  advancing  portion  of  the  belt  runs  on 
nearer  to  the  base. 

If  a  similar  cone  is  so  placed  that  its  base  coincides  with  that 
of  the  first  one,  when  the  centre  line  of  the  belt 
has  mounted  to  the  common  base  it  will  remain  in 
that  position,  as  any  displacement  from  such  position 
-o  would  bring  about  the  condition  tending  to  return 
it.  Pulleys  are,  therefore,  usually  made  "  crowning" 
to  keep  the  belt  on  the  centre.  If  the  pulley  is 
crowned  about  -J  inch  for  each  foot  in  width,  the  belt 
will  ordinarily  evince  no  tendency  to  run  off,  provi- 
ded the  axes  of  the  connecting  shafts  are  parallel. 
If  the  shafts  are  out  of  alignment,  the  belt  tends 
to  run  toward  the  edges  at  which  the  belt  is  tightest, 
unless  the  shafts  are  very  much  "  out." 
It  is  frequently  desirable  to  stop  the  driven  shaft  without  stop- 
ping the  driver,  and  a  common  method  of  doing  this  is  by  means 
of  "  tight-and-loose  "  pulleys.  Two  pulleys  are  placed  side  by  side 
on  the  driven  shaft,  one  of  which  is  fastened  to  the  shaft,  while  the 
other  is  free  to  rotate  relative  to  this  shaft,  but  is  prevented  by 
collars  from  moving  axially.  The  hub  of  the  tight  pulley  usually 
serves  as  one  of  these  collars,  and  the  rims  should  not  quite  touch. 
A  pulley  is  secured  on  the  driving-shaft,  opposite  the  tight-and-loose 
pulley,  having  a  width  (or  face)  equal  to  the  combined  width  of 
both  of  the  latter.  A  belt  of  about  the  width  of  either  of  the  single 
pulleys  connects  one  of  them  and  the  wide-faced  driving  pulley. 
When  this  belt  is  on  the  tight  pulley,  the  follower  is  driven;  but  if 
it  is  shifted  to  the  loose  pulley  the  follower  will  stop,  although  the 
belt  continues  to  run.  The  belt  is  easily  shifted  by  applying  a 
lateral  pressure  to  the  advancing  edge,  as  explained  in  Art.  123. 
It  is  usual  with  tight-and-loose  pulleys  to  make  them  both  crown- 
ing, so  that  the  belt  will  remain  upon  either  when  it  is  shifted;  but 
to  facilitate  shifting  the  wide  driving  pulley  is  generally  made  with 
a  straight  face  (cylindrical  surface). 

126.  Length  Of  Belt— The  length  of  belt  is  usually  determined 


WRAPPING-  CONNECTORS.  225 

by  direct  measurement  if  the  pulleys  are  constructed  and  mounted, 
or  by  measuring  a  drawing  if  the  work  is  not  built  and  erected. 
This  length  may  be  calculated  for  either  an  open  or  a  crossed  belt 
(Figs.  198  and  199,  respectively).  This  calculation  is  seldom  of 
practical  value  simply  for  the  determination  of  the  length,  but  it 
plays  an  important  part  in  the  correct  design  of  "stepped-cone 
pulleys,"  such  as  are  used  on  the  countershafts  and  spindles  of 
lathes  and  other  machines  for  securing  changes  of  speed.  The 
importance  of  this  calculation  will  appear  from  the  discussion  of 
the  next  article. 

The  open  band  of  Fig.  198  causes  the  follower  to  rotate  in  the 
same  direction  as  the  driver,  while  the  crossed  band  (Fig.  199) 
gives  the  follower  a  rotation  in  an  opposite  direction.  This  will 
be  seen  to  agree  with  the  general  statement,  of  Art.  33;  for  with 
the  open  belt  both  fixed  centres  are  on  the  same  side  of  the  line  of 
action  (the  driving  side  of  the  belt)  ;  while,  with  the  crossed  belt 
these  centres  are  on  opposite  'sides  of  the  line  of  action.  Owing  to 
the  rubbing  of  the  sides  of  the  belt  where  they  cross,  the  open 
band  is  used  when  it  is  feasible.  The  crossed  band  has  the  advan- 
tage of  a  larger  arc  of  contact,  which  has  an  important  effect  on  the 
adhesion,  especially  on  the  smaller  pulley;  but  with  wide,  stiff  belts, 
particularly  when  the  distance  between  centres  is  small,  the  warp- 
ing of  the  belt  may  largely  destroy  this  advantage. 

It  is  evident  that  the  length  of  belt  is  different  in  the  two  cases, 
other  conditions  being  the  same.  The  following  are  the  algebraic 
expressions  for  the  length  of  belts: 

The  angle  MQT  =  mqt  =  SqQ  =  0. 

For  crossed  belts  (Fig.  199), 


sin  0  =  ,  and  Tt  =  qS  =  V7F  —  (R 

n  _ 

pen  belts,  sin  0  =  —  ^ 
The  length  of  the  crossed  belt 


n  _  __ 

For  open  belts,  sin  0  =  —  —  ,  and  Tt  =  qS  =  V  d*  —  (R  —  r)*. 


*=  L  =  2  v'<TJ-(#  +  r)'  +  7rR+  2#sin-'-+7rr-f  2r  sin'1 

«  2  V  d*  -  (R  +  r)'+  (R  +  r)  (*  +  2  sin->^-p).     ...     (1) 


226  KINEMATICS   OF  MACHINERY. 

The  length  of  the  open  belt 


-r)2sm->--.      «-   .     .     (2) 

It  follows  from  (1  )  that  a  crossed  belt  which  is  of  proper  length 
for  any  pair  of  pulleys,  R  and  r,  will  be  of  correct  length  for  any 
other  pair  of  pulleys,  R'  and  r'  (on  the  same  shafts)  if  R  -{-  r  = 
R'  -f  r',  that  is,  if  the  sum  of  the  radii  is  constant;  for  (R  -f  r)  is 
the  only  variable  quantity. 

It  will  be  seen,  however,  in  (2)  that  if  R'  -f  r'  =  R  .+  r  ; 
72'  —  r'  cannot  equal  7?  —  r,  unless  R'  =  R  and  r'  =  r. 

An  open  belt  of  the  correct  length  for  two  pulleys,  R  and  r,  on 
fixed  shafts  would  not,  therefore,  be  of  exactly  the  right  length 
for  another  pair  of  pulleys,  R'  and  r',  on  these  same  shafts,  if 
R'  -f-  r'  =  R  +  r,  unless  the  two  larger  pulleys  are  equal,  and  the 
two  smaller  pulleys  are  also  equal.  Such  a  belt  might  be  made  to 
run  if  the  distance  between  shafts  were  quite  great  and  the  change 
in  sizes  of  pulleys  were  small  ;  but  it  would  not  be  equally  tight  on 
the  different  sets. 

126.  Stepped  Cones.—  It  is  often  important  to  change  the  speed 
of  a  machine  which  is  driven  from  a  shaft  having  uniform  speed. 
Cones,  as  shown  in  Fig.  204,  might  be  placed  upon  the  counter- 
shaft  and  on  the  spindle  of  the  machine.  If  a  crossed  belt  is  used, 
it  would  be  equally  tight  at  all  corresponding  positions  on  these 
cones,  but  an  open  belt  would  not  be;  and  in  order  to  have  it  so, 
"swelled  "  cones,  as  shown  (exaggerated)  in  Fig.  205,  would  be  re- 
quired. Such  conical  drums  have  the  advantage  of  permitting 
every  possible  variation  in  speed  within  limits;  but  the  belt  tends 
to  mount  towards  the  large  ends  of  both,  which  increases  the  strain 
upon  the  belts  and  the  pressure  upon  the  bearings. 

The  stepped  cones,  Fig.  206,  are  more  compact  than  conical 
drums,  and  they  avoid  the  objection  just  mentioned.  It  follows 
from  the  preceding  discussion  that  for  a  crossed  belt  the  sum  of 


WRAPPING-CONNECTORS. 


227 


the  radii  of  any  mating  pair  of  steps  should  be  a  constant.  But 
the  sum  of  the  radii  of  the  intermediate  pairs  of  steps  should  be 
greater  than  the  sum  for  the  outside  steps  when  using  an  open 
belt.  Rankine's  Machinery  and  Millwork  gives  a  method  of  deter- 
mining the  swell  of  the  cones  (Fig.  205)  from  which  the  radii  of 
the  intermediate  steps  of  a  stepped  cone  can  be  derived.  A  much 


Fig.  204. 


Fig.  205. 


Fig.  206. 


more  convenient  approximate  graphical  method  is  described  by  Mr. 
€.  A.  Smith,  in  the  Trans,  of  the  A.  S.  M.  E.,  Vol.  X,  page  269. 

Lay  off  Qq  (Fig.  207)  equal  to  the  distance  between  shafts;  draw 
the  circles  with  radii  R  and  r,  equal  to  the  radii  of  the  known  pul- 
leys; at  C,  half  way  between  q  and  Q,  erect  the  perpendicular 
CG  =  .SHQq,  and  with  G  as  a  centre,  draw  the  arc  mm  tangent  to 
iT.  The  belt  line  of  any  other  pair  of  steps  should  be  tangent  to 
•mm.  R'  and  rr  are  radii  of  two  such  steps;  and  the  velocity  ratio 
when  using  these  ^teps  will  be  R'  -r-  r'  =  FQ  -f-  Fq.  Let  Qq  =  d\ 
let  Fq  =  x;  and  call  the  desired  ratio  a. 


Now 


=  a. 


x  = 


a-  r 

value  of  x\  draw  FT'  tangent  to  mm* 

respective  centres,  and  tangent  to  FT',  give  the  required  wheels  with 


Lay  off  Fq  equal  to  this 
Circles  with  Q  and  q  as  the 


228 


KINEMATICS    OF    MACHINERY. 


radii  R'  and  rf.     This  method,  as  here  outlined,  only  applies  when 
the  belt  angle,  $  of  Fig.  207,  is  less  than  about  18°. 

Tl 


The  original  paper,  referred  to  above,  gives  a  modified  method 
for  use  when  <fi  is  greater  than  18°. 

An  even  more  convenient  method  has  been  recently  devised 
by  Dr.  L.  Burmester  of  Leipzig,  Germany.     At  an  angle  of  45° 

with  the  horizontal  line  A B 
(Fig.  208),  draw  the  line  AC 
=  d  =  the  distance  between 
the  centres  of  the  shafts. 

Take  CD=—,   perpendicular 

2 

to  AC  at  C.  With  A  as  a 
centre  draw  a  circular  arc 
passing  through  D.  On  this 
arc  locate  E,  so  that  the 
vertical  distance  between  E 
and  a  point  F,  on  AC,  is 
EF  =  R— r  =  the  difference 
between  the  radii  of  the  given 
pair  of  steps.  Extend  EF 

to  G,  making  FG=r.  Through  G  draw  the  horizontal  OG, 
intersecting  AC  at  0.  Then  EG  =  R,  and  OG=r.  Draw  OE. 
Let  the  angle  EOG=d.  Then  tan  6  =  EG  +  OG=R  +  r  =  n  =  the 
given  velocity  ratio.  The  radii  R^  and  rl  for  any  other  velocity 
ratio  nx  are  found  as  follows:  Through  0  draw  OEt  at  such  an 


Fig.  208. 


WRA  PPIXG-COXXECTORS. 


229 


angle  0t  with  OG  that  tan  O^n^,  intersecting  the  circular  arc 
at  J£j.  Draw  the  vertical  E^  intersecting  OG  produced  at  Gt. 
Then  Eft^R^  and  OG^  =  rr 

To  secure  satisfactory  results  the  above  construction  should 
be  accurately  made  to  as  large  a  scale  as  may  be  convenient. 
The  values  obtained  should  be  checked  by  calculating  the  exact 
belt  length  for  each  pair  of  steps,  using  equation  (3)  of  the  pre- 
ceding article. 

127.  Twisted  Belts. — It  is  sometimes  desired  to  connect  two 
shafts  which  are  not  parallel  by  a  belt.     This  can  often  be  done, 
by  the  use  of  a  twisted  belt.     (See  Fig.  209.)     Suppose  two  pulleys 
in  the  plane  of  the  paper  (the  lower  one  shown 

by  the  dotted  circle)  to  be  on  parallel  shafts,  Q 
and  q,  Fig.  209. 

Draw  Tt  tangent  to  each  pulley  at  the  centre 
of  its  face,  and  on  the  side  at  which  the  belt 
leaves  it.  Then,  if  the  lower  pulley  and  its 
shaft  be  turned  about  Tt  as  an  axis  to  the  posi- 
tion shown  by  the  full  lines,  the  planes  of  the 
two  pulleys  will  intersect  in  Tt.  The  line,  A  a, 
in  which  the  belt  advances  upon  the  lower 
pulley  will  lie  in  the  plane  of  this  pulley.  The 
line,  bB,  in  which  the  belt  advances  upon  the 
upper  pulley,  will  also  lie  in  its  plane.  It  has 
been  shown  (Art.  123)  that  the  direction  of  the 
receding  side  of  the  belt  does  not  affect  the  ac- 
tion; therefore  this  belt  will  remain  upon  the  pulleys  and  con- 
tinue to  drive.  If  the  motion  of  the  pulleys  be  reversed,  however, 
the  belt  will  at  once  run  off,  because  its  advancing  side  does  not  lie 
in  the  plane  of  the  pulley  under  this  new  condition.  If  the  angle 
through  which  the  lower  shaft,  q'q',  is  turned  is  90°,  the  term 
quarter-turn  belt  is  applied. 

128.  Guide-pulleys. — The  only  condition  necessary  in  order 
that  a  belt  shall  run  on  a  pulley  is  that  the  centre  line  of  its  advanc- 
ing side  shall  lie  in  the  central  plane  of  the  pulley.     By  use  of 


230 


KINEMATICS    OF   MACHINERY. 


guide-pulleys,  or  idlers,  two  shafts  either  intersecting  at  any 
angle  or  not  in  one  plane  can  be  connected  by  a  belt.  If  desired, 
the  belts  can  be  made  to  run  in  either  direction  by  so  placing  guide 
pulleys  that  both  sides  of  the  belt  lie  in  the  planes  of  the  pulleys. 
Fig.  210  shows  a  few  of  the  possible  applications  of  guide-pulleys 
in  connecting  shafts  which  are  not  parallel.  In  the  arrangement 
of  Figs.  210  (a)  and  210  (c)  the  belt  may  run  in  either  direction;  but 


Fig.  210. 


in  Fig.  210  (b)  the  belt  will  only  remain  on  the  pulleys  when  it  is 
run  in  the  direction  indicated  by  the  arrows. 

129.  Belt-tighteners. — It  is  sometimes  desirable  to  provide  for 
variation  in  distance  between  shafts,  to  secure  a  greater  arc  of  belt 
contact,  to  take  up  stretch  of  belt,  or  to  avoid  the  use  of  clutches 
and  tight-and-loose  pulleys,  by  employing  a  belt-tightener.  This 
is  simply  an  idle  pulley,  mounted  on  a  suitable  frame  in  such  a 
way  that  it  can  be  moved  by  screws,  levers,  weights,  or  springs,  to 
change  or  maintain  the  tension  of  the  belt.  The  only  condition 
necessarily  complied  with  is  that  the  centre  line  of  the  advancing 


WRAPPING-CONNECTORS. 


231 


Fig.  211. 


side  of  the  belt  shall  lie  in  the  central  plane  of  the  pulley  to  which 
it  runs. 

130.  Sprocket-wheels  for  Chains. — One  form  of  transmission- 
chain  and  sprocket-wheel  is  shown  in  Fig.  211.  The  true  pitch 
line  of  the  wheel  is  a  closed  equal- 
sided  polygon,  each  side  being 
equal  to  the  length  of  a  link  from 
centre  to  centre  of  the  pins.  Or 
if  a  circle  be  drawn  about  Q  pass- 
ing through  the  centres  of  all  the 
pins  that  lie  on  the  wheel,  the 
centre  lines  of  the  corresponding 
links  form  chords  of  this  circle. 
As  each  link  approaches  or  recedes 
from  the  wheel,  one  of  its  pin 
centres  rotates,  relative  to  the 

wheel,  about  the  other  pin  centre,  describing  a  circular  arc  relative 
to  the  wheel.  Thus,  Fig.  211,6  describes  the  arc  bp  relative  to  the 
wheel  as  the  link  ab  wraps  upon  the  wheel.  In  order  that  the  teeth 
of  the  wheel  shall  allow  the  links  to  drop  smoothly  into  place,  the 
actual  tooth  outline  may  be  an  arc  parallel  to  pb,  as  shown  by  the 
arc  mn.  Adjacent  sides  of  two  teeth  may  be  joined  by  an  arc  about 
/?,  the  radius  of  which  is  equal  to  the  radius  of  the  pin,  or  bushing, 
which  joins  the  links.  By  making  the  outer  portions  of  the  teeth 
lie  somewhat  inside  the  arcs  inn,  the  pin  does  not  rub  upon  the 
tooth  as  it  approaches  the  wheel,  but  it  will  fall  into  place  and 
reach  a  bearing  at  the  end  of  its  approaching  action.  The  backs 
of  the  teeth  are  sometimes  relieved  more  than  the  fronts  or  driv- 
ing sides  when  the  rotation  is  to  be  in  one  direction  only.  Since 
the  true  pitch  line  of  the  wheel  is  a  polygon  instead  of  a  true 
circle,  the  velocity  ratio  is  not  exactly  constant  with  sprocket- 
wheels.  The  irregularity  is  usually  not  important  with  wheels  of 
a  considerable  number  of  teeth. 

If  two  sprocket-wheels  are  connected  by  a  chain,  their  angular 
velocity  ratio  is  inversely  as  their  numbers  of  teeth,  as  in  toothed 


232 


KINEMATICS  OF  MACHINERY. 


gearing.  This  is  a  more  convenient  measure  of  the  velocity  ratio 
than  the  radii  of  the  pitch  circles,  or  the  circles  inscribing  the  pitch 
polygons. 

Modifications  of  the  construction  shown  in  Fig.  211  permit  the 
employment  of  chains  with  various  forms  of  links,  or  of  the  special 
chains  called  "  link-belts/'  etc. 

A  wheel  frequently  used  in  cranes  for  the  common  chain,  with 
oval  links  of  round  iron,  is  shown 
by  Fig.  212.  Every  other  link 
lies  on  the  wheel  with  its  plane  in 
the  central  plane  of  the  wheel ;  while 
the  intermediate  links  lie  in  planes 
normal  to  these.  Pockets,  as  shown, 
prevent  slipping,  and  the  flanges 
at  the  siies  strengthen  the  projec- 
ing  teeth  greatly,  so  that  there  is  no 
difficulty  in  getting  a  wheel  stronger 
than  the  chain  itself.  Fig.  212. 

131.  Wrapping-connectors  with  Varying  Angular  Velocity 
Ratio. — As  already  shown,  flexible  connectors  can  be  used  to  trans- 
mit a  variable  angular  velocity  ratio,  for  instance,  by  using  such 

forms  as  are  shown  in  Figs.  54,  55, 
and  56.  A  somewhat  different  appli- 
cation is  shown  in  Fig.  213.  It  has 
been  employed  in  chronometers  and 
F|9'  2I3>  watches  to  secure  a  more  uniform 

driving  action  to  the  mechanism  as  the  spring  runs  down.  The 
spring  is  placed  in  the  cylindrical  piece,  called  the  barrel,  and  as  it 
uncoils  the  small  chain  is  wound  upon  the  barrel  and  unwound 
from  the  conical  piece,  called  a  "fusee."  It  will  be  seen  that  as 
the  spring  runs  down  the  pull  on  the  connector  diminishes,  but  the 
"leverage"  of  the  connector  upon  the  follower  is  increased  corre- 
spondingly, and,  therefore,  the  driving  effort  transmitted  to  the 
mechanism  may  be  kept  quite  uniform. 


CHAPTER  VIII. 
TRAINS    OF    MECHANISM. 

132.  Substitution  of  a   Train  for  a  Simple  Mechanism. — It  is 

kinematically  possible  to  transmit  motion  between  two  parallel 
shafts  with  any  required  angular  velocity  ratio  by  either  a  single 
pair  of  gears  or  of  pulleys;  but  there  are  practical  conditions 
which  often  make  it  desirable  to  effect  the  required  transmission 
of  motion  by  a  series  of  mechanisms,  or  a  compound  mechanism, 
instead  of  by  a  single  pair  of  gears,  of  pulleys,  etc.  Such  an  arrange- 
ment constitutes  a  train  of  mechanism.  The  train  may  contain 
pulleys  with  belts,  ropes,  chains,  gears,  screws,  and  linkwork,  any 
or  all;  and  it  may  be  used  to  transmit  motion  between  other 
members  than  parallel  shafts.  If  two  shafts  are  to  be  connected 
by  gears,  and  the  required  velocity  ratio  is  high,  the  difference  in 
the  size  of  the  gears  may  be  inconveniently  great  if  a  single  pair  is 
used.  That  is,  the  large  wheel  may  occupy  too  much  room,  or  be 
difficult  to  swing,  or  the  small  gear  may  have  so  few  teeth  that  it 
would  be  objectionable.  For  example :  suppose  the  velocity  ratio 
is  25  to  1,  and  that  strength  requires  wheels  of  2  (diametral)  pitch. 
Then  if  the  pinion  be  given  only  12  teeth,  it  will  be  6  inches  in 
diameter,  and  the  large  wheel  will  be  25  X  6  —  150  inches  in 
diameter  (  =  12£  feet).  Now,  suppose  that  an  intermediate  shaft 
be  introduced.  This  intermediate  shaft  can  be  connected  to  the 
slower  of  the  original  shafts  by  using  a  pair  of  gears  which  will  cause 
it  to  rotate  5  times  to  1  rotation  of  this  primary  shaft,  and  it  can 
be  connected  to  the  faster  of  the  original  shafts  by  a  pair  of  gears 
which  will  give  it  1  rotation  to  5  of  the  latter  shaft;  then  as  each 

233 


234  KINEMATICS  OF  MACHINERY. 

revolution  of  the  first  shaft  corresponds  to  5  revolutions  of  this 
intermediate  shaft,  and  as  each  of  its  revolutions  corresponds  to  5 
revolutions  of  the  last  main  shaft,  it  is  evident  that  the  velocit  7 
ratio  between  the  first  and  last  shaft  is  5  X  5  to  1,  equal  25  to  1  - 
as  required. 

The  velocity  ratio  of  first  shaft  to  intermediate  and  of  interme- 
diate to  last  shaft  are  not  necessarily  equal.  They  may  be  any- 
thing whatever  if  the  product  of  the  separate  angular  velocity 
ratios  equals  the  required  ratio  between  the  first  and  the  last  shaft. 
Furthermore,  the  three  axes  need  not  lie  in  one  plane;  that  is,  the 
centres  need  not  be  in  one  straight  line.  It  is  thus  seen  that  the 
use  of  a  train  in  place  of  a  simple  mechanism  permits  considerable 
flexibility  in  the  arrangement;  this  will  be  clearly  seen  from  an 
examination  of  various  actual  trains. 

In  a  similar  manner  to  that  of  the  preceding  illustration,  an  in- 
termediate shaft  may  be  used  in  a  belt  transmission  when  the 
velocity  ratio  is  high.  Such  an  arrangement  is  frequently  seen 
when  a  slow-speed  engine  drives  a  dynamo.  The  engine  is  belted 
to  a  "  jack-shaft/'  which  in  turn  drives  the  dynamo.  This  may 
be  desirable  either  to  avoid  an  excessively  large  pulley  or  to  avoid 
an  extremely  wide  angle  between  the  sides  of  the  belt.  The  effect 
of  a  large  belt  angle  is  to  reduce  the  arc  of  contact  on  the  smaller 
pulley;  this  reduces  the  adhesion  of  the  belt  and  increases  liability 
of  slip  of  the  belt. 

Other  considerations  than  a  high  velocity  ratio  may  make  it 
desirable  to  substitute  a  train  for  a  simple  mechanism;  for  instance, 
to  secure  a  required  directional  relation,  for  compactness,  etc.  A 
familiar  example  of  such  a  train  is  seen  in  the  back-gear  mechanism 
(Fig.  217),  as  used  on  lathes  and  other  machine  tools. 

A  shaft  which  carries  intermediate  gears  of  a  train  may  itself 
drive  some  member  which  requires  a  motion  different  from  that  of 
the  last  member.  Thus,  in  clockwork,  the  gear  on  the  shaft  to 
which  the  minute-hand  is  fixed  drives  the  hour-hand  through  a 
reducing  pair  of  gears,  and  it  may  also  drive  a  second  hand  at  a 
higher  rate. 


TRAINS  OF  MECHANISM. 


235 


133.  Value  of  a  Train. — Suppose  four  axes,  I,  II,  III,  and  IV 
(Fig.  214)  to  be  arranged  as  shown  and  connected  by  toothed 
gears  of  which  the  circles  «,  b,  c, 
etc.,  are  the  pitch  lines.  The 
wheel  a  meshes  with  b;  c  meshes 
with  d,  and  e  meshes  with  f.  Both 
of  the  wheels  b  and  c  are  secured 
to  the  shaft  II;  hence  they  must 
rotate  as  one  piece,  having  the 
same  angular  velocity  at  any  in- 
stant. Likewise,  d  and  e  are 
both  secured  so  shaft  III,  and  they  have  the  same  angular  velocity. 
Let  the  angular  velocities  of  the  shafts  I,  II,  III,  and  IV  be  repre- 
sented by  al9  a^  <*3,  and  a^  respectively.  Two  gears  which  mesh 
together  must  have  the  same  pitch ;  hence  the  numbers  of  teeth 
are  proportional  to  the  circumferences,  to  the  diameters,  or  to  the 
radii.  But  their  angular  velocities  are  inversely  as  the  radii,  and 
therefore  inversely  as  the  numbers  of  teeth  on  the  wheel.  It 
follows  that  if  a,  b,  c,  etc.,  are  the  numbers  of  teeth  on  the  wheels 
designated  by  these  letters,  that. 


214. 


a, 

=  —  '  X 


(Xa 


<X. 


an 


a. 


a  .  c .  e 


In  this  train  a  is  the  driver  and  b  is  the  follower  in  the  first  pair; 
c  is  the  driver  and  d  the  follower  in  the  second  pair;  and  e  is  the 
driver  and  /  is  the  follower  in  the  third  pair.  It  will  be  seen  from 
the  above  expression  for  al  -=-  or4  that  the  angular  velocity  ratio  of 
the  first  driving-shaft  I  to  the  last  driven  shaft  IV  equals  the  con- 
tinued product  of  the  numbers  of  teeth  in  the  driven  wheels  di- 
vided by  the  continued  product  of  the  numbers  of  teeth  in  th^ 
driving  wheels.  The  angular  velocity  ratio  between  two  wheels  i 
the  direct  ratio  of  the  numbers  of  revolutions  they  make  in  a  unit 


236  KINEMATICS  OF  MACHINERY. 

of  time,  as  a  minute.     In  finding  the  value  of  a  train,  any  of  the 

factors,  —  L,  —  ,  etc.,  may  be  expressed  in  terms  of  the  numbers  of 
tf-t    a3 

teeth,  radii,  diameters,  or  revolutions  per  unit  of  time  of  the  pair 
of  wheels  involved;  but  if  the  latter  relation  is  used  the  ratio  is 
direct,  while  with  the  other  terms  the  inverse  ratio  is  to  be  taken. 
It  is  not  necessary  that  these  different  factors  be  all  given  in  the 
same  terms.  Thus  if  a  has  60  teeth  and  b  has  16  teeth;  c  is  24 
inches  in  diameter,  and  d  is  8  inches  in  diameter;  e  makes  75  revo- 
lutions and  f  makes  250  revolutions  per  minute, 

75         4 


For  every  revolution  of  I,  IV  makes  37-J  revolutions;  hence  if  I 
makes  10  revolutions  per  minute,  IV  will  make  375  revolutions  per 
minute. 

A  train  is  shown  by  Fig.  215  in  which  the  shaft  I  drives  the 
shaft  II  through  pulleys  connected  with  a  crossed  belt  ;  III  is 
driven  from  II  by  an  open  belt;  and  IV  is  driven  from  III  by 
gears.  An  expression  similar  to  that  given  above  can  be  used  to 


~'  'Fig.  215. 

find  the  ratio  of  the  angular  velocities  of  I  to  IV.  Thus  suppose 
that  the  pulleys  a,  b,  c,  and  d  are,  respectively,  8,  20,  10,  and  24 
inches  in  diameter;  and  that  the  gears  e  and  /  have  18  and  70  teeth 
respectively;  then 

<xl  _  20       24       70  _  70 
51  "  8   X  10  X  18  -  F 


TRAINS  OF  MECHANISM.  237 

The  shaft  I  makes  70  revolutions  to  3  of  the  shaft  IV  (or  23^ 
to  1).  Or,  if  I  makes  175  revolutions  per  minute,  IV  makes 

175  X  3 

—  —  —  =  7-J  revolutions  per  minute. 
<0 

In  general,  if  there  are  m  shafts  connected  by  gears  or  pulleys, 
the  angular  velocity  ratio  of  the  first  shaft  to  the  last  is 

• 

(x\  of,       of,       a,          ctm-1 

—  =  _i  X  —'  X  ~  .  .  .  -=-*  ......      (2) 

am  a^       a,       at  <xm 

x''' 

If  the  numbers  of  revolutions  per  unit  of  time  of  the  first  and 
last  shaft  are  JV,  and  Nm,  respectively,  Ni  :  Nm  :  :  «i  :  am  (as  the 
angular  velocity  of  a  member  is  proportional  to  its  revolutions  per 
unit  of  time)  ;  hence 


(3) 


Belt  connections  are  usually  preferred  when  the  speeds  of  the 
shafts  are  high,  the  distance  between  centres  is  great,  and  a  moderate; 
amount  of  slipping  is  not  seriously  objectionable.  When  the  speed 
is  slow,  the  distance  between  shafts  is  comparatively  small,  or  when 


positive  transmission  is  essential,  gears  are  better.  When  this  last 
condition  is  not  a  requisite,  and  the  distance  between  shafts  is  too 
small  to  use  belting  advantageously,  frictional  gears  are  occasionally 
employed.  When  the  distance  between  two  shafts  is  very  great, 
rope  transmission  (wire  or  hemp)  may  be  used. 

A  train  is  shown  in  Fig.   216  in  which  the  axes  are  not  all 


238  KINEMATICS  OF  MACHINERY. 

parallel.  A  pinion  a  on  shaft  I  drives  the  spur-gear  I  on  II;  a 
pair  of  bevel-gears  c  and  d  connect  II  and  III,  and  a  worm  e  on  III 
drives  the  worm-wheel  f  on  IV.  If  the  numbers  of  teeth  on  «,  b,  c, 
d,  e,  and  /are  15,  45,  25,  35,  1,  and  50,  respectively, 

ai       45       35       50 


or  the  first  shaft  makes  210  revolutions  to  every  revolution  of  the 
last  shaft. 

It  will  be  seen  that  the  expression  for  the  value  of  a  train,  as 
deduced  above,  is  general,  and  applies  to  all  cases  when  the  proper 
substitutions  are  made. 

134.  Directional  Relation  in  a  Train.— When  two  spur-gears 
mesh  together  they  rotate  in  opposite  direction  ;  hence,  if  the  train 
is  made  up  entirely  of  spur-gears  the  adjacent  axes  rotate  in  oppo- 
site directions,  and  the  alternate  axes  (first,  third,  fifth,  etc.,  or 
second,  fourth,  etc.)  have  rotations  in  the  same  direction.  If  such 
a  train  has  an  odd  number  of  axes  the  first  and  last  axes  will 
Rotate  in  the  same  direction;  while  if  there  is  an  even  number 
of  axes  the  first  and  last  will  rotate  in  opposite  directions.  Thus 
in  the  train  of  Fig.  214  the  shafts  I  and  IV  rotate  in  opposite 
directions. 

If  one  of  the  gears  is  an  internal  (or  annular)  gear  the  shaft 
to  which  it  is  attached  rotates  in  the  same  direction  as  the  pinion 
which  meshes  with  this  gear. 

If  an  open  belt  connects  the  pulleys  on  two  shafts  these 
shafts  rotate  in  the  same  direction,  while  a  crossed  belt  connect- 
ing two  shafts  causes  them  to  rotate  in  opposite  directions.  Thus 
in  Fig.  215,  I  and  II  rotate  in  opposite  directions  ;  II  and  III 
rotate  in  the  same  direction,  and  III  and  IV  rotate  in  opposite 
directions.  In  this  example  there  is  an  even  number  of  shafts,  but 
there  is  one  open-belt  connection  ;  hence,  the  rotations  of  the  first 
and  last  shaft  are  in  the  same  direction,  as  will  appear  from  an  in- 
spection of  the  figure. 

135.  Back  Gears. — The  common  screw-cutting  lathe  and  many 


TRAINS  OF  MECHANISM. 


239 


other  machine  tools  have  a  gear-train  through  which  the  stepped 

cone  can  be  connected  with  the 

spindle.     This  is  shown    in  Fig. 

217.    The  cone  A  is  driven  by  a 

belt  from  another  cone   on   the 

counter-shaft.     When   the   back 

gears  are  thrown  out  and  the  cone 

of  the  headstock  is  locked  to  the 

spindle  C,  these  two  members  (the 

cone  and  spindle)   move  as  one 

I  iece.    If  the  cone  has  three  steps 

the  spindle  can  be  given  three  different  speeds  from  the  uniformly 

revolving  countershaft.     By  means  of  the  back  gears  the  number 

of  speeds  of  the  spindle  is  doubled  without  adding  more  steps  to 

the  cone.    When  the  back  gears  are  "in"  the  cone  is  not  secured 

directly  to  the  spindle,  but  is  free  to  rotate  upon  it.     A  pinion,  « , 

attached  to  the  cone,  engages  with  the  first  back  gear,  6,  which  is 

mounted  on  the  shaft  B.     This  shaft  has  another  gear,  c,  secured  to 

the  opposite  end ;  and  c  engages  with  the  gear  d,  which  is  attached 

to  the  spindle.    The  angular  velocity  of  the  cone  may  be  designated 

by  0:1;  that  of  the  two  back  gears  by  <*2,  and  that  of  the  spindle  by 

<x3;  then  the  angular  velocity  ratio  of  the  cone  to  the  spindle  is 

—  —  — X— ;  or— •  —  the  product  of  the  numbers  of  teeth  on  b 
<xt      a2     a3         «3 

and  d  divided  by  the  product  of  the  numbers  of  teeth  on  a  and  c. 

The  cones  on  both  the  spindle  and  the  countershaft  are  com- 
monly equal  with  engine  lathes;  but  on  wood  lathes  (which  do  not 
use  back  gears)  the  countershaft  cone  is  usually  the  larger,  to  secure 
the  requisite  high  speed  of  the  spindle  from  a  moderate  speed  of 
countershaft. 

A  countershaft  runs  at  90  revolutions  per  minute,  the  four 
steps  of  the  (equal)  cones  are  12",  9£",  7",  and  4|"  in  diameter; 
the  numbers  of  teeth  on  the  gears  a,  6,  c,  and  dare  28,  100,  24,  and 
88,  respectively.  The  following  speeds  of  the  spindle  may  be  obtained : 
Direct  driving  (back  gears  out) : 

Belt  on  largest  step  of  countershaft  cone  and  smallest  step  of 
spindle  cone: 


240 


KINEMATICS  OF  MACHINERY. 


-jo 


(1)  Spindle  speed  =  90  X—  = 

4.o 


240. 


Belt  on  next  smaller  step  of  countershaft  cone  and  next  larger 
step  of  spindle  cone: 

(2)  Spindle  speed  =  90  X  ^  =  122.14. 

Belt  on  next  pair  of  steps  : 

7 

(3)  Spindle  speed  =  90  X  r-=  =  66.32. 

9.5 

Belt  on  smallest  countershaft  step  and  largest  spindle  cone 
step: 

(4)  Spindle  speed  =  90  X^  =  33.76. 

Driving  through  back  gears:     Value  of  back-gear  train: 
~  =  1QU  x  "88  =  jji  =  -076    (nearly),  giving   four    speeds   with 
back  gears  which  may  be  found  by  multiplying  the  four  speeds  as 
calculated  above  by  —  ;  or 

(*! 

(5)  =  240  X  .076  =  18.24. 

(6)  =  122.14  X  .076  =  9.4  —  . 

(7)  =  66.32  X  .076  =  5.0  +. 

(8)  =33.76  X  .076  =  2.56+. 

The  student  should  take  the  necessary  data  from  an  actual 
lathe  and  compute  the  various  spindle  speeds. 

136.  The  Idler.  —  It  was  shown  in  Art.  134  that  one  result  of  an 

intermediate  shaft  in  a  spur-gear 
train  is  to  affect  the  direction 
of  rotation  between  the  first  "and 
third  shafts.  If  these  two  shafts 
were  connected  directly  by  a  pnir 
of  spur-gears  they  would  rotate 
in  opposite  directions  ;  but  when 


Fig.  218. 


connected  through  an  intermediate  shaft  they  rotate  in  the  aame 
direction. 


TRAINS  OF  MECHANISM.  241 

In  Fig.  218  three  shafts,  I,  II,  and  III,  are  shown  connected 
"in  series"  by  the  gears  a,  b,  and  c.  If  these  letters  designate  the 
numbers  of  teeth  on  the  corresponding  wheels  : 

«! b      cx.2 c  «i b       c  _     c 

•  j •      dllCl.  P\     "~7_~"    ~~~    ~"~*~~3 

a2      a      a3      o  «3      a       b        a 

or  the  intermediate  wheel  does  not  affect  the  ratio  between  the 
angular  velocities  of  the  first  and  third  shafts,  but  it  does  cause  them 
to  rotate  in  the  same  direction. 

Such  an  intermediate  wheel  in  a  train  is  called  an  idler.  That 
the  idler  does  not  affect  the  ratio  between  the  times  of  revolu- 
tions of  a  and  c  can  be  seen  directly  by  inspection,  for  the  linear 
velocity  of  a  point  in  the  pitch  circle  of  a  must  equal  that  of  a 
point  in  the  pitch  circle  of  b,  and  also  points  in  the  pitch  circles  of 
b  and  c  must  have  the  same  linear  velocities  ;  therefore,  as  all 
points  in  the  pitch  circle  of  b  have  the  same  velocity,  the  linear  ve- 
locity of  points  in  the  pitch  circles  of  a  and  b  are  the  same,  and  the 
angular  velocities  of  these  two  wheels  are  inversely  as  their  radii, 
just  as  if  they  engaged  directly. 

Fig.  219  shows  the  "  tumbling-gears  "usually  placed  in  the  head- 
stock  of  the  screw-cutting  lathe  to  enable  the  operator  to  easily 
change  the  direction  of  feed,  or  to 
cut  either  a  right-  or  a  left-handed 
screw.  The  gear  a  is  connected 
to  the  lathe-spindle  and  d  is  on 
the  stud  through  which  the  feed- 
rod  or  lead  screw  is  driven.  In 
the  position  shown,  a  drives  b,  b 
drives  c,  and  c  drives  </.  It  will  be 
seen  that  a  and  d  rotate  in  opposite  directions,  and  as  b  and  c  are 
both  idlers,  the  action  is  equivalent  to  direct  engagement  of  a  and  d. 
The  gears  b  and  c  are  carried  on  a  support  which  can  be  swung 
about  the  centre  of  d  by  a  suitable  handle  extending  through  the 
front  of  the  head  stock,  and  when  this  handle  is  dropped  down,  c 
can  be  meshed  directly  with  a,  b  being  thrown  out  of  mesh  with  *. 


242  KINEMATICS  OF  MACHINERY. 

In  this  position  I  simply  rotates,  as  it  remains  in  mesh  with  c ;  but 
a  drives  c  directly,  and  c  drives  d.  There  are  but  three  axes  in  the 
train  in  this  condition  ;  hence  a  and  d  rotate  in  the  same  direction. 
137.  The  Screw-cutting  Train.— In  the  screw-cutting  lathe  a 
long  screw,  called  the  lead  screw,  or  leading  screw,  is  placed  par- 
allel to  the  bed,  and  the  carriage  which  holds  the  lathe  tool  may  be 
connected  to  this  screw  by  a  clamp-nut.  When  this  nut  is  closed 
upon  the  screw  the  carriage  will  be  fed  along  the  bed  as  the  screw  is 
turned.  If  the  screw  has  four  threads  to  the  inch  (J-inch  pitch), 

e  every  turn  of  the  screw  will  feed  the 
tool  J  inch  parallel  to  the  axis  of  the 
lathe.  If  the  screw  has  the  gear  g 
(Fig.  220)  mounted  upon  it  at  one 
end,  the  screw  will  make  one  revolu- 
tion for  each  revolution  of  this  gear. 

Fig.  220.  '    u  Now   suppose   the   gear  e  to   rotate 

with  the  lathe-spindle  ;  then  if  e  is  equal  to  g,  and  is  connected 
with  it  by  the  idler  /,  eack  revolution  of  the  spindle  compels  the 
screw  to  make  one  revolution.  If  a  cylindrical  piece  of  stock  is 
mounted  in  the  lathe  so  that  it  rotates  with  the  spindle,  and  a  thread 
tool  in  the  tool-post  is  fed  (transversely)  till  it  enters  this  cylin- 
drical piece,  it  will  be  seen  that  the  feed-mechanism  will  cause  the 
tool  to  cut  a  thread  on  the  stock  which  is  a  reproduction  (as  to 
pitch)  of  the  leading  screw  ;  for  the  tool  has  a  longitudinal  motion 
of  \  inch  for  each  revolution  of  the  work,  and  a  proportional  motion 
for  any  fraction  of  a  revolution.  The  idler,/,  is  carried  on  a  slotted 
piece  which  can  be  swung  about  the  axis  of  the  screw,  VI,  and  the 
stud  upon  which /rotates  can  be  set  at  different  distances  from  VI, 
along  the  radial  slot.  The  gear  g  could  then  be  replaced  by  one 
of  a  different  size,  /  could  be  moved  along  to  engage  with  it,  and 
by  swinging  the  support  of /it  could  also  be  made  to  engage  with  e, 
in  which  position  it  can  be  clamped.  By  this  means  the  velocity 
ratio  between  the  spindle  and  the  gear  can  be  varied. 

Suppose  it  is  desired  to  cut  a  screw  of  8  threads  to  the  inch 
(J"  pitch).     By  placing  a  gear  (g)  twice  as  large  as  e  on  the  screw, 


TRAINS  OF  MECHANISM.  243 

each  revolution  of  the  spindle  will  cause  g  to  make  but  half  a 
revolution,  and  the  tool  will  be  fed  only  half  the  pitch  of  the  lead- 
ing screw  along  the  stock  during  one  complete  revolution  of  the 
latter.  To  cut  a  screw  of  6  threads  per  inch,  g  must  be  1J  (f )  the 
size  of  e,  then  a  revolution  of  the  spindle  and  of  the  stock  would 
occur  for  |  of  a  revolution  of  the  screw;  or  the  feed  per  revolution 
of  the  spindle  would  be  J-  X  f  =  ^  of  an  inch.  It  will  appear  that 
screws  of  different  pitches  may  be  cut  from  a  given  lead  screw,  each 
of  which  is,  in  a  sense,  a  reproduction,  reduced  or  enlarged,  of  this 
screw. 

The  screw-cutting  lathe  is  provided  with  a  set  of  gears  to  be 
used  as  indicated  above,  for  cutting  all  the  whole  number  (even) 
threads  throughout  a  rather  wide  range.  Such  a  set  is  called  a  set 
of  change  gears. 

A  typical  arrangement  is  a  combination  of  the  trains  shown  by 
Figs.  219  and  220.  The  gear  a  (Fig.  219)  is  on  the  spindle,  and 
it  drives  d  in  either  direction,  through  the  tumblers,  as  explained 
in  the  preceding  article.  The  stud  (IV)  to  which  d  is  attached 
passes  through  the  end  of  the  headstock  and  e  (Fig.  220)  is  fastened 
upon  its  outer  end.  Then,  by  means  of  the  change-gears  any  re- 
quired thread  within  the  range  of  the  lathe  can  be  cut,  either  right- 
or  left-handed. 

The  gear  on  the  outer  end  of  the  stud  may  be  fixed,  g  only 
being  changed;  but  provision  is  usually  made  for  changing  either  e 
or  g  (or  both).  Sometimes  a  and  d  are  not  equal  (d  being  usually 
twice  as  large  as  a  in  such  cases) ;  then  the  ratio  between  e  and  g 
must  be  taken  accordingly.  More  often,  however,  a  and  d  are 
equal. 

A  certain  lathe  of  16"  swing  has  a  lead  screw  of  4  threads  per 
inch,  and  change  gears  of  the  following  numbers  of  teeth  : 
24,  30,  36,  42,  48,  48,  54,  60,  66,  69,  72,  78,  84.  With  the  24  gear 
on  the  stud  it  will  cut:  5,  6,  7,  8,  9,  10,  11,  11 J,  12,  13,  and  14 
tli reads  per  inch,  with  the  following  gears,  respectively,  on  the 
screw  :  30,  36,  42,  48,  54,  60,  66,  69,  72,  78,  84. 

The  11J  thread  corresponds  to  a  standard  pipe  thread,  and  it  is 


244  KINEMATICS  OF  MACHINERY. 

consequently  convenient  to  be  able  to  cut  this  pitch  in  a  lathe. 
To  permit  cutting  this  thread  in  the  lathe,  it  is  now  not  uncommon 
to  provide  a  gear  for  it.  It  will  be  noticed  that  the  above  list  of 
change-gears  includes  two  48-tooth  gears.  These  are  used  for  cut- 
ting a  4-thread  screw,  one  of  them  being  placed  on  the  stud  and  the 
other  on  the  screw.  For  cutting  2  (or  3)  threads,  one  of  the  48 
tooth  gears  is  put  on  the  stud,  and  the  24  (or  36)  gear  must  be 
used  on  the  screw;  as  the  screw  must  make  2  (or  !•£)  revolutions, 
as  the  case  may  be,  for  each  revolution  of  the  spindle. 

When  the  stud-gear  e  makes  the  same  number  of  revolutions 
as  the  spindle,  the  following  formula  may  be  used  for  finding  the 
change-gears,  in  which  e  equals  the  teeth  in  the  stud-gear,  g 
equals  the  teeth  in  the  screw-gear,  t  equals  the  threads  per  inch 
of  the  lead  screw,  and  n  equals  the  threads  per  inch  to  be  cut  : — 

—  =  — .      If  the  stud-gear  is  fixed,  g=  -e.      Any   two    gears   of 

t  6  t 

the  set  may  be  taken  which  have  numbers  of  teeth  in  the  ratio  of 
n  to  t.  If  the  stud-gear  e  does  not  make  the  same  number  of  revolu- 
tions as  the  spindle,  that  is,  if  a  and  d  of  Fig.  219  are  not  equal, 

—  =  —  X  — .     The  idlers,  b,  c9  and /do  not  enter  into  the  calcula- 
t       a        & 

tions,  for  they  do  not  affect  the  velocity  ratio. 

The  above  screw-cutting  train  is  given  as  an  example  of  the 
ordinary  arrangement;  but  among  lathes  of  various  makes  there  are 
many  modifications  in  detail  to  be  found.  All  ordinary  screw- 
cutting  lathes  have  a  mechanism  which  is  fundamentally  that  given 
above. 

It  will  be  noticed  that  in  the  series  of  gears  given  above  there 
is  a  constant  difference  of  6  teeth  between  the  successive  gears 
(neglecting  the  gear  for  11  \  threads  and  the  extra  48-tooth  gear). 
In  any  such  system,  for  whole  numbers  of  threads  to  the  inch,  this 
constant  difference  equals  the  number  of  teeth  on  the  stud-gear 
divided  by  the  threads  per  inch  of  the  lead  screw  (  =  e  ~-  ^),  when 
the  spindle  and  stud  have  the  same  number  of  revolutions  per  unit 
of  time.  If  the  stud  is  geared  to  make  only  one  revolution  to  two 


TRAINS  OF  MECHANISM.  245 

revolutions  of  the  spindle,  the  difference  between  successive  wheels 
is  half  of  that  given  by  this  rule. 

The  change-gears  should  always  be  constructed  on  the  involute 
system,  as  this  is  the  only  system  in  which  the  centre  distances  can 
vary  without  affecting  the  constancy  of  the  velocity  ratio. 

Many  lathes  are  provided  with  a  screw-cutting  train  which 
-can  be  "compounded."  In  this  arrangement  the  simple  idler/ 
(Fig.  220)  is  replaced  by  two  gears  of  different  diameters,  secured 
together  and  rotating  on  the  stud  V  as  one  piece.  The  gear  e 
meshes  with  one  of  these  intermediate  gears,  and  the  gear  g  (which 
must  be  correspondingly  displaced  laterally  along  its  axis,  VI) 
meshes  with  the  other.  This  pair  of  intermediate  gears  (unlike 
the  idler/)  affects  the  velocity  ratio  between  the  spindle  and  the 
screw,  because  of  the  difference  in  the  diameters  of  the  two  inter- 
mediate gears.  The  velocity  ratio  as  found  by  the  preceding 
method  must  be  multiplied  by  the  ratio  of  the  two  intermediate 
gears.  This  latter  ratio  is  usually  2  to  1,  or  1  to  2,  depending 
upon  whether  the  larger  of  the  compounding-gears  engages  with  e 
or  with  g. 

138.  Epicyclic  Trains. — It  was  shown  in  Art.  39  that  any  mem- 
ber of  a  linkage  could  be  considered  as  the  fixed  link,  and  appar- 
ently different  mechanisms  would  be  thus  obtained.  This  is  true 
of  gear-trains  as  well  as  of  linkages.  If  one  of  the  gears  of  a  gear- 
train  is  made  the  fixed  member,  instead  of  the  bar  supporting  the 
gears,  the  mechanism  is  called  an  epicyclic  train,  because  in  its 
action  one  or  more  of  the  gears  rotates  on  its  axis  at  the  same  time 
that  it  revolves  about  the  axis  of  the  fixed  gear,  so  that  points  in 
it  describe  epicycloidal  curves. 

Generally  in  these  mechanisms  the  angular  velocity  ratio 
between  the  last  rotating  gear  and  the  arm  which  carries  it  is  re- 
quired. In  Fig.  221  let  a  and  b  be  two  gears  mounted  on  an  arm  c, 
so  that  if  c  were  fixed,  a  and  b  would  form  a  simple  gear-train.  Now 
suppose  that  a  is  made  fast  to  some  fixed  body,  so  that  it  really 
becomes  the  fixed  member  of  the  train.  Then  c  can  rotate  around 
O,  carrying  b  with  it,  b  itself  rotating  relative  to  c  around  its 


246 


KINEMATICS   OF  MACHINERY. 


axis  at  0' .  It  is  required  to  find  the  number  of  revolutions  that 
1)  will  make  around  its  own  axis,  relative  to  the  fixed  member,  for 
every  revolution  of  c  around  0. 

First,  let  a  be  disconnected  from  the  fixed  body,  so  that  a,  b, 
and  c  can  make  one  revolution  as  one  piece  in  the  direction  indi- 
cated around  0.  Then  b  will  make  one  revolution  around  0',  solely 
because  of  its  motion  around  0.  This  can  be  seen  by  noting  the 


Fig.  221. 


positions  of  any  point,  as  P,  relative  to  0'  during  different  phases 
of  the  revolution,  as  shown.  Now  if  c  is  held  stationary  and  a  is 
rotated  backward  one  revolution,  6  will  occupy  the  position  it 
would  have  had  if  a  had  been  held  stationary  all  the  time.  Let  r 
be  the  angular  velocity  ratio  between  b  and  a  when  c  is  held  sta- 
tionary. Then,  when  a  is  rotated  backward  one  revolution,  b 
must  receive  r  turns  forward,  and  the  total  number  of  revolutions 
which  6  will  make  for  one  revolution  of  c  around  o  =  n=  1  +r,  and 
its  direction  of  rotation  will  be  the  same  as  that  of  c.  It  is  evident 
•that  c  can  be  rotated  in  either  direction,  and  the  result  obtained 
above  will  still  hold. 

If  we  place  an  idler  between  a  and  6  (Fig.  222) ,  or  if  a  is  an 
annular  gear  (Fig.  223),  the  direction  of  motion  of  6  is  reversed 
so  that  it  will  make  one  turn  in  the  direction  of  rotation  of  c  and 


TRAINS  OF  MECHANISM. 


247 


minus  r  turns  in  the  opposite  direction,  or  n  =  1  —  r.  The  rota- 
tion of  b  (Fig.  222) ,  relative  to  the  fixed  member,  may  or  may  not 
be  in  the  same  direction  as  that  of  c,  depending  on  the  value  of  r. 


Fig.  222.  Fig.  223. 

A  special  case  is  that  when  r— 1,  whence  n  =  o,  and  b  does  not 
rotate  around  0',  relative  to  the  fixed  member,  but  has  a  simple 
motion  of  circular  translation.  The  direction  of  rotation  of  b 


•  224-  I        Fig.  225. 

(Fig.  223)  must  always  be  opposite  to  that  of  r,  since  r  can  not 
be  less  than  I  when  a  is  an  annular  gear. 

In  general,  if  the  first  and  last  gear  of  the  train  turn  in  opposite 
directions  relative  to  the  supporting  bar,  the  last  gear  makes  1  +  r 


248  KINEMATICS  OF  MACHINERY. 

revolutions  in  the  same  direction  as  the  bar  for  each  revolution  of 
that  member;  when  the  first  and  last  gears  turn  in  the  same 
direction  relative  to  the  supporting  bar  the  last  gear  makes  1  —  r 
revolutions  for  each  revolution  of  the  bar.  When  1  —  r  is  positive, 
the  last  gear  and  the  bar  turn  in  the  same  direction,  but  when  1  —  r 
is  negative  they  turn  in  opposite  directions. 

It  is  evident  that  a  compound  train  can  be  used  between  a  and 
6,  as  shown  in  Fig.  224;  and  the  results  will  be  the  same  as  with 
the  arrangement  shown  in  Fig.  222,  since  we  are  concerned  only 
with  the  angular  velocity  ratio  of  b  to  a  and  the  direction  of  rota- 
tion of  b  relative  to  a,  regardless  of  how  these  are  obtained. 

Further,  the  axes  need  not  lie  in  a  straight  line,  but  can  occupy 
any  position  relative  to  each  other  as  long  as  the  gears  rnesh  prop- 
erly. A  common  arrangement  is  that  shown  in  plan  and  elevation 
in  Fig.  225,  where  the  axis  of  b  is  made  to  coincide  with  that  of  the 
fixed  gear  a.  The  gears  d  and  e  are  fast  together,  and  are  carried 
by  c,  which  is  free  to  rotate  on  the  spindle  /.  It  is,  therefore,  a 
compound  chain,  having  three  axes,  and  is  called  a  reverted  train. 

In  this  form  it  is  used  extensively  for  obtaining  great  velocity 
ratios  between  the  arm  c  and  the  lust  gear  b. 

For  example  : 

Let  a  have  99  teeth 
«  b   "  100  " 
"  d   "  101  " 
"  e  "  100  " 

Then  99  X  101  _  9999  m 

~  100  x  100  ~~  Toooo' 

and   since   a  and  b  rotate  in  the  same  direction  relative  to  c, 
—          rev.  ,  or  c  must  make  10,000  revolutions 


in  order  to  make  b  rotate  once,. 

The  application  of  epicyclic  gears  to  hoisting  devices  will  be 
obvious  from  the  above.  They  are  also  used  for  feed-mechanisms 
on  large  boring-bars,  in  machines  for  making  wire  ropes,  etc.,  etc. 


PROBLEMS  AND  EXEECISES. 


NOTE. — A  large  number  of  exercises  on  Kinematics  have  been  arranged  by 
Mr.  A.  T.  Bruegel,  formerly  of  Sibley  College,  Cornell  University,  who  has  kindly 
consented  to  the  use  of  some  of  them  in  the  present  work. 

The  original  set  contains  three  classes  of  exercises,  intended: — to  illustrate 
the  principles  treated;  to  drill  the  student  on  the  application  of  these  principles 
in  the  solution  of  definite  problems,  and  to  extend  the  range  of  the  text.  The 
exercises  given  below  were  selected  mainly  from  those  of  the  second  class,  and 
they  include  a  few  additional  ones  by  the  writer. 

The  references  in  brackets  are  to  the  articles  in  the  text  which  relate  most 
directly  to  the  particular  problem. 

J.  H.  B. 

1.  [Art.  2.]    A  train  has  attained  a  speed  of  112  miles  per  hour  for  a 
short  distance.    Express  its  velocity  in  feet  per  minute,  in  feet  per  second, 
and  in  inches  per  second. 

2.  [Art.  2.]    The  stroke  of  an  engine  is  18",  and  the  crank-pin  makes 
250  revs,  per  minute.      Express  the  linear  velocity,  or  rate  of  motion,  of 
this  pin  in  feet  per  minute  ;  in  inches  per  second  ;  in  feet  per  second. 

3.  [Art.  4.]    The  drivers  of  a  locomotive  are  5  feet  in  diameter,  and  the 
Stroke  of  the  piston  is  24  inches.     Calculate  the  mean,  or  average,  piston 
speed  (linear  velocity)  in.  feet  per  minute  when  the  locomotive  runs  at  the 
rate  of  40  miles  per  hour. 

4.  [Art.  4.]    An  engine  with  a  stroke  of  5  feet  makes  65  revs,  per  min. 
What  is  the  mean  piston  speed  ? 

5.  [Arts.  4,  5,  6.]    A  train  runs  110  miles  in  2  hours  and  40  minutes. 
Drivers,  64  inches  in  diameter.     Stroke  of  piston,  22  inches.     Required  : 

(a)  Mean  velocity  of  engine,  in  feet  per  minute,  relative  to  the  earth. 
(6)  Mean  velocity  of  piston  relative  to  engine-frame. 

(c)  Mean  velocity  of  crank-pin  relative  to  engine-frame. 

(d)  Mean  velocity  ratio  between  piston  and  crank-pin. 

249 


250  KINEMATICS  OF  MACHINERY. 

(e)  Mean  velocity  of  point  in  tread,  relative  to  frame. 
(/)  Path  of  point  in  tread  relative  to  frame. 
(g)  Path  of  point  in  tread  relative  to  earth. 
(h)  Kind  of  motion  of  crank-pin  and  piston. 

6.  [Art.  14.]  Represent,  graphically,  the  mean  velocity  of  the  crank-pin 
of  Prob.  5  (c).     Use  scale  of  1000  feet  per  minute  to  the  inch. 

7.  [Art.  14.]  Represent,  graphically,  mean  velocity  of  piston  in  Prob.  5r 
(6).     Scale  700  feet  per  min.  to  the  inch. 

8.  [Art.  17.]  An  engine  with  stroke  of  18  inches  makes  220  revolutions 
per  minute.     Find,  graphically,  the  vertical  and  horizontal  components  of 
the  crank-pin  velocity  when  the  crank  makes  angles  of  30°,  120°,  and  210° 
respectively,  with  its  initial  position  on  centre  line  of  engine.     Write  the 
results  in  feet  per  second  upon  the  lines  which  represent  them. 

9.  [Art.  17.]    A  resultant  pv  (Fig.  18)  equals  70  feet  per  second  ;   the 
components  pvi  and  pvi  equal  64  feet  and  48  feet  per  second,  respectively. 
Find,  graphically,  the  directions  of  the  components.     Two  solutions- are 
possible. 

10.  [Art.  17.]  A  velocity  of  450  feet  per  minute  is  to  be  resolved  inta 
two  components  making  angles  with  it,  on  opposite  sides,  of  30  and  60 
degrees,  respectively. 

11.  [Art.  17.]  Three  component  motions  in  one  plane  have  velocities  of 
60,  80,  and  100  feet  per  minute,  respectively;  the  first  is  vertically  upward;, 
the  second  makes  an  angle  of  30  degrees  to  the  right  with  it  ;  and  the  third 
an  angle  of  45  degrees  with  the  second,  also  to  the  right.   Find  the  value  of 
the  resultant,  graphically. 

12.  [Art.  17 J  A  point  moving  upward  and  to  the  right,  at  an  angle  of 
60  degrees  with  the  horizontal,  has  a  velocity  of  40  feet  per  minute. 

(a)  Resolve  it  into  a  vertical  and  a  horizontal  component. 

(&)  Resolve  it  into  two  components,  one  of  which  makes  an  angle  of  45 
degrees  with  the  horizontal  towards  the  right  and  has  a  velocity  of  30  feet 
per  minute. 

(c)  Resolve  it  into  two  components  of  25  and  50  feet  per  minute,  re- 
spectively. Graphical  solutions  required. 

13.  [Art.  17.]  An  engine  of  24  inches  stroke  makes  160  revolutions  per 
minute.     The  connecting-rod  is  four  times  the  length  of  the  crank.     Find 
(graphically)  the  rate  of  motion  of  the  cross-head  when  the  crank  is  at  45 
degrees  and  at  90  degrees  with  the  centre  line  of  engine. 

14.  [Art.  18.]  A  locomotive  running  at  the  rate  of  35  miles  per  hour 
has  63-inch  driving-wheels  and  24-inch  stroke.     Find  the  linear  and  the 
angular  velocity  of   the  crank-pin   relative  to  the  frame.     Give   results 
in  feet  per  minute  and  in  inches  per  second. 

15.  [Art.  18.]  An  engine  makes  600  strokes  per  minute.     Fly-wheel  is- 


PROBLEMS  AND  EXERCISES.  251 

on  the  crank-shaft.  Find  the  angular  velocity  of  the  fly-wheel,  the  linear 
velocity  of  a  point  3  feet  from  the  centre  of  the  shaft,  and  also  of  a  point 
4  inches  from  the  centre.  Express  results  in  feet  per  minute. 

When  is  the  angular  velocity  of  a  point  expressed  by  a  number  greater 
than  that  of  the  linear  velocity  ? 

16.  [Art.  18.]  A  wheel  10  feet  in  diameter  makes  100  revolutions  per 
minute.     What  are  the  linear  and  the  angular  velocity  of  a  point  in  the  rim  ; 
of  a  point  6  inches  from  the  axis  ;  of  a  point  12  inches  from  the  axis  ? 
Give  all  the  results  in  feet  per  second. 

17.  [Art.  18.]  (a)  A  body  moves  in  a  straight  line  with  a  linear  velocity 
of  25  feet  per  second.     What  is  its  angular  velocity  ? 

(b)  A  governor-ball  is  8  inches  from  the  axis  of  rotation  when  revolv- 
ing at  the  rate  of  300  revolutions  per  minute.  Express  its  linear  and  its 
angular  velocity  in  units  of  feet  and  minutes  and  in  inches  and  seconds. 

18.  [Art.  19.]  Locate  all  the  instant  centres  for  the  mechanism  of  Prob. 
13,  at  the  phases  specified. 

19.  [Art.  20.]  Same  engine  as  Prob.  13,  pressure  on  piston  taken  at 
10,000  Ibs.     Draw  the  parallelograms  of  the  forces  acting  upon  the  crank- 
pin  and  which  constrain  it  to  move  in  a  prescribed  path.     Make  a  separate 
sketch  for  each  of  the  following  phases,  the  crank  rotating  clockwise  0  = 
45°,  150°,  210°,  300°,  and  the  two  positions  at  which  the  crank  makes  a 
right  angle  with  the  connecting-rod.      [0  is  the  angle  which  the  crank 
makes  with  the  centre  line  of  the  engine.] 

Also  state  whether  the  connecting-rod  and  crank  are  under  tensile  0** 
compression  stresses  at  each  of  the  above  positions. 

20.  [Art.  30.]  An  arm  12  inches  long,  rotating  uniformly  at  30  rev.  per 
minute,  drives  an  arm  30  inches  long  through  an  intermediate  link  36 
inches  in  length  ;  distance  between  fixed  centres  48  inches.  Find,  by 
method  of  instant  centres,  velocity  of  ro\ lower  when  driver-pin  is  on 
the  line  of  and  between  the  fixed  centres,  and  90,  180,  and  270  degrees 
ahead  of  this  position  (4  phases).  Also  state  the  directional  relation  in 
each  case. 

Express  velocities  in  feet  per  minute  and  tabulate  results.  Graphical 
solution. 

21.  [Art.  40.]  Prove  that,  in  the  mechanism  of  Fig.  74,  Oac  must  lie  in 
the  intersection  of  the  lines  b  and  d  (prolonged). 

22.  [Art.  41.]  Draw  velocity  diagram  for  cross-head  of  an  engine  hav- 
ing stroke  of  16",  and  connecting-rod  40"  long.     Engine  makes  150  revs, 
per  min.     Prove  for  one  ordinate.    Also  construct  velocity  diagram  of 
cross- head  with  a  connecting-rod  48"  long. 

23.  [Art.  41.]  Fig.  77;  a  =  6",  c  =  14",  b  =  18",  d  =  20",  and  a  makes 


252  KINEMATICS  OF  MACHINERY. 

30  revs,  per  min.     Construct  velocity  diagram  for  point  Obc\  (a)  upon  its 
path  as  a  base;  (b)  upon  a  rectilinear  base.     Prove  for  one  ordinate. 

24.  [Art.  43.]  Draw  the  pair  of  rolling  centrodes  for  the  relative  motion 
of  the  cross-head  and  crank  (Fig  .70).     Also  draw  the  pair  of  centrodes  for 
the  relative  motion  of  the  connecting-rod  and  frame. 

25.  [Art.  46. J  Two  shafts  are  6  inches  apart;  driver  makes  50  rev.  per 
min.     Construct  a  pair  of  rolling  ellipses  for  connecting  the  shafts,  such 
that  follower  shall  have  a  maximum  rate  of  75  rev.  per  min.     What  is  the 
minimum   rate  of  follower  ?    Give   major  and   minor  axes  of    ellipses. 
(Draw  pitch  lines  one-half  size.) 

26.  [Art.  46.]  Distance  between  fixed  centres  (opposite  foci  of  ellipses) 
is  8".     Construct  two  rolling  elliptical  arcs,  such  that  the  velocity  ratio 
will  vary  between  the  limits;  2:  3,  and  4:  3,  for  an  angular  motion  of  the 
driver  of  60  degrees. 

27.  [Art.  48.]    Fig.  90.     Take  0  ...(/  =  5",  and   Op  =  H".     Draw 
Ap  perpendicular  to  Op,  and  construct  the  curve  which  will  roll  upon  Ap\ 
Ap  and  this  curve  to  rotate  about  the  fixed  centres  0  and  0',  respectively. 

28.  [Art.  51.]  Two  parallel  shafts,  24"  between  centres,  are  to  be  con- 
nected by  rolling  cylinders.     One  shaft  is   required   to  make   350   revs. 
Clockwise  ;  while  the  other  makes  500  revs,  counter-clockwise.     What  are 
the  proper  diameters  ? 

29.  [Art.  51.]  Same  data  as  Prob.  29,  except  that  the  shafts  are  both  to 
turn  in  the  same  direction.     Required,  the  diameters. 

30.  [Art.  51.]  Design  rolling  conical  frusta  to  transmit  motion  between 
two  shafts  which  intersect   at  an  angle  of  60  degrees.     Driver  to  make 
800  rev.  to  400  rev.  of  follower.    How  may  directional  relation  be  changed 
Without  affecting  the  velocity  ratio  ? 

31.  [Art.  55.]  A  pair  of  grooved  friction- wheels  have  pitch  diameters 
of   8   feet  and  2  feet ;   working  depth  of  groove  equals  1£  inches.     The 
pinion  makes  180  rev.  per  min.     Find  maximum  sliding  action,  in  feet  and 
in  inches  per  min.;  assuming  no  slip  at  the  pitch  lines. 

32.  [Art.  62.]  Epicycloidal  gearing.   Data  : — pitch  diameter  of  driver  = 
12";   of   follower  =  8";   1   diametral  pitch ;   addendum  length   of  large 
wheel  =  1";   of  small  wheel  =  f";   backlash  =  0  ;  bottom   clearance  = 
0.1";  ratio   of   arc   of  approach   to  arc  of  recess  =  f  ;  arc  of  action  = 
circular  pitch. 

Required  : — Diameters  of  describing  circles;  full  construction  of  three 
teeth  of  each  wheel ;  angles  of  maximum  obliquity  during  both  approach 
and  recess. 

33.  [Art.  63.]  Epicycloidal  gearing.    Data :— Pitch  diameters  14"  ana 
10";  diameter  of  describing  circles  equal  to  radius  of  smaller  wheel ;  back- 


PROBLEMS  AND  EXERCISES.  253 

lash  =  clearance  =  TV;  angles  of  approach  and  recess  equal;  H"  circular 
pitch. 

Required  : — Least  addenda  which  will  insure  contact  between  two 
pairs  of  teeth  at  all  times  ;  angle  of  action  in  terms  of  the  pitch.  Test 
accuracy  of  the  construction  by  rolling  a  tracing  of  one  set  of  teeih  upon 
the  other. 

34.  [Art.  68,  69.]  Annular  involute  gears.     Construct  several  teeth  of 
annular  gear  and  pinion  complying  with  the  following  conditions  : 

Diameters  of  pitch  circles  12"  and  20";  2  diametral  pitch;  clearance  = 
TV  ;  backlash  =  0  ;  addendum  =  \"  ;  root  =  addendum  +  clearance  ; 
profiles  to  be  involutes  line  of  action  at  75°  with  line  of  centres),  as  far  as 
possible;  roots  of  pinion  to  be  radial  inside  its  base  circle,  outline  of 
annular  wheel  teeth  to  be  continued  from  the  proper  point  by  hypocloid  of 
suitable  form.  Mark  the  point  where  this  hypocloid  joins  the  involute. 

35.  [Art.  77.]  Approximate  tooth   outline.     Data : — Circular  pitch  = 
4";  number  of  teeth  =  18  ;  diameter  of  describing  circle  =  radius  of  12- 
tooth  pinion;  addendum  =  33  pitch  ;  root  =  .37  pitch. 

Required  : — (a)  Construction  of  tooth  outline  by  the  exact  method  ;  (6) 
approximate  (circular  arc)  outlines  by  the  Willis,  Grant,  aud  Unwin 
methods,  for  comparison.  All  of  these  outlines  should  pass  through  the 
same  point  on  the  pitch  circle,  and  should  be  very  carefully  drawn  with 
fine  lines. 

36.  [Art.  77.]  Draw  outline  of  an  involute  tooth  for  a  wheel  18"  diam. 
with  27  teeth.    Compare  this  with  Grant's  approximation  for  involute  teeth, 
by  method  similar  to  that  outlined  in  Prob.  36. 

37.  [Art  79.]  Design  a  mitre-gear  (one  of  a  pair  of  equal  bevel-gears) 
with  greatest  pitch  diameter  =  10"  ;  20  teeth,  epicycloidal  outlines  ;  length 
of  teeth  along  the  elements  equal  to  2|  times  the  circular  pitch,  and  other 
dimensions  with  customary  proportions.     Thickness  of  rim  equal  to  roots 
of  teeth.     Draw  two  views  of  one  quarter  of  the  wheel. 

38.  [Art.  90.]   A  No.  5  Brown  &  Sharpe  cutter  is  used  for  involute 
wheels  having  from  21  to  25  teeth.     Construct,  accurately,  the  outline  for 
one  tooth  of  an  involute  gear  of  1  diametral  pitch,  21  teeth  ;  then  with  the 
same  pitch  and  pitch  points  draw  the  outline  for  a  wheel  of  25  teeth.    This 
comparison  will  show  double  the  necessary  maximum  error  in  using  one 
cutter  through  this  range. 

39.  [Art.  90.]  A  No.  3  B.  and  S.  cutter  is  used  for  wheels  having  35  to 
54  teeth.     Make  a  construction  (1  diametral  pitch)  for  one  tooth  of  each  of 
these  extreme  sizes  of  wheels,  and  compare  the  difference  with  that  found 
in  Prob.  38. 

40.  [Art.  90.]  Compare  the  maximum  error  in  using  an  "  M  "  cutter  for 


254  KINEMATICS  OF  MACHINERY. 

epicycloidal  gears  of  27  and  29  teeth,  with  the  error  in  cutting  50  and  59 
teeth  by  an  "  R  "  cutter.     Use  3"  circular  pitch. 

41.  [Art.  94.]     Construct  a  cam  on  a  base  circle  3"  diam.,  to  make  one 
revolution  per  minute,  and  to  impart  to  a  roll  1"  diam.,  whose  straight  line 
of  motion  passes  through  the  centre  of  the  axis,  a  stroke  of  2".     The  roll  is 
to  rise  uniformly  during  25  seconds,  remain  at  rest  for  20  seconds,  and  descend 
during  the  remainder  of  the  revolution  with  a  uniformly  accelerated  motion 
(spaces  passed  over  in  equal  times  in  the  ratio  of  1,  3,  5,  7,  etc.,  as  in  falling 
bodies)  . 

42.  [Art.  94.]     Draw  a  cam  which  by  oscillating  through  an  angle  of  60° 
shall  give  a  uniformly  ascending  and  descending  motion  to  a  sliding-bar 
the  line  of  motion  of  which  passes  4"  to  the  right   of  the  axis.     Stroke  of 
bar  =3";    base-circle  =  10".     Cam  acting  on  a  roll  1£"  diam.  at  end  of  the 
bar. 

43.  [Art.  94.]    The  follower  of  a  cam  is  a  rocker,  22  inches  long  (roll  2" 
diam.  at  free  end),  with  fixed  centre  4  inches  above  and  24  inches  to  the 
right   of  cam-shaft.     Lowest  position   of  follower  is   horizontal  and   cam 
rotates  uniformly,  moving  follower  through  30  degrees.     During  90  degrees 
of  rotation  of  cam  follower  describes  angles  in  the  ratio  of  1,  3,  5,  3,  2,  and 
I,  and  then  rests  during  the  next  90  degrees  of  rotation  of  cam,  and  descends 
with  uniform  angular  velocity  during  the  remainder  of  rotation  of  cam. 

44.  [Art.  95.]     A  cam  is  to  act  upon  a  straight  tangential  follower,  with 
The  working  face  of  the  latter  perpendicular  to  its  line  of  motion.     (See  Fig. 
142.)     The  follower  is  to  be  moved  uniformly  upward,  a  total  distance  of 
iy,  while  the  cam  rotates   through   120°;    then  follower  is  to  rest  during 
an  angular  motion  of  the  cam  of  90°;    and  to  descend  with  a  uniformly 
accelerated  motion  during  the  completion  of  the  rotation.     Make  base  circle 
of  cam  =  4". 

45.  [Art.  100.]    Design  a  worm  and  wheel  such  that 


Required:   T,  t,  D,  d,  <j>. 

Give  the  teeth  the  involute  rack-and-pinion  outline  at  middle  section, 
and  mark  contact  points  of  teeth. 

46.  [Art.   104.]     If,  in  the  four-link  chain  of  Fig.  159,  a  =  3";    6  =  4"; 
d=10";    find  the  limits  between  which  the  length  of  c  must  lie  in  order 
to  permit  continuous  rotation  of  a. 

47.  [Art.  104.]     Taking  same  data  as  Prob.  46,  is  it  possible  to  give  c 
such  a  length  that  a  drag-link  mechanism  results;    that  is,  so  that  both  a 
and  6  shall  rotate  continuously?  Test  this  by  finding  the  limiting  values  of 
c  for  the  drag-link  chain,  with  given  values  of  a,  b,  and  d. 

48.  [Art.    104.]    If,  in  the  drag-link  chain  of  Fig.  160,  a  =  7";    6  =  6"; 
d=3";    find  the  limiting  values  of  c  which  will  permit  continuous  rotation 
of  both  a  and  6. 


PROBLEMS   AND  EXERCISES.  255 

49.  [Art.  106.]     (See  Fig.  164.)     An  engine  with  a  stroke  of  18"  has  a 
connecting-rod  45"  long. 

(a)  Calculate  the  distance  of  the  piston  (or  cross-head)  from  the  end  of 
stroke  (a-c)  when  the  crank  angle  (6  measured  from  A)  is  60°. 

(6)  Calculate  the  distance  from  the  other  end  of  the  stroke  when  6  =  120°. 

(c)  Calculate  the  distance  from  cross-head  to  middle  of  the  stroke  (q — ra), 
when  6  =  90°,  or  270. 

50.  [Art.  106.]     With  data  as  in  Prob.  50,  except  that  connecting-rod  is 
54"  long,  calculate  (a),  (6),  (c). 

51.  [Art.  106.]     Data  as  in  Prob.  49.     Calculate  crank  angles  at  which 
velocity  of  cross-head  (piston)  equals  velocity  of  crank-pin.     Also  find  ratio 
of  piston  velocity  to  crank-pin  velocity  when  crank  and  connecting-rod  form 
a  right  angle  at  C. 

52.  [Art.  110.]     (Fig.  173.)     The  perpendicular  distance  from  Q  to  the 
line  of  stroke,  h — g,  is  2";   radius  of  a  crank  =  3";   crank  makes  20  rev.  per 
minute;   connecting-rod  C  — c  =  9".     Find  length  of  stroke  of  c;   and  con- 
struct velocity  diagram  of  c  for  forward  and  return  strokes  on  a — b  as  a 


53.  [Art.  113.]     (Fig.  175.)     Design  a  Whitworth  quick-return  mechanism 
such  that  length  of  stroke  c  shall  be  10";   ratio  of  times  of  forward  and  re- 
turn strokes  =  2  :1;  rqadius  of  driving-crank  (OP)  =  4";  length  of  connecting- 
rod  =12". 

Construct  velocity  diagram  of  c  for  both  strokes. 

54.  [Art.  115.]     (Fig.  179.)     The  stroke  of  a  beam-engine  is  4  feet;  dis- 
tance from  line  of  piston  motion  to  beam  centre  (d)  =  5  feet.     Find  proper 
length  of  beam  for  minimum  obliquity  of  connecting-rod. 

55.  [Art.    126.]     A   countershaft   runs   at    100   rev.   per  minute.     This 
countershaft  is  to  drive  a  spindle  through  stepped  cones  and  an  open  belt 
at  150,   100,  or  75  rev.  per  minute.     Largest  step  on  countershaft  =  14" 
diam.     Distance  between  centres  =  7  feet.     Find,  graphically,  the  diameters 
of  all  the  steps.     Check  the  accuracy  of   the  method  by  calculating  the 
lengths  of  belts  for  each  of  the  three  pairs  of  steps. 

56.  [Art.    133.]     (See   Fig.    215.)     The   diameter   of   a  =  24";     6  =  40"; 
c=36";   d=54";   e  has  15  teeth;   and  /  has  48  teeth.    Find  velocity  ratio 
and  the  directional  relation  between  a  and/. 

57.  [Art.   133.]     (Fig.  214.)     Data:— a  has  60  teeth;    6  has  16  teeth; 
diam.  of  c  =  24";   diam.  of  d  =  8";   e  makes  75  rev.  per  min.   and/  250  rev. 
per  min.     How  many  rev.  per  min.  does  a  make;  and  what  is  the  directional 
relation  between  a  and  /? 

58.  [Art.  133.]     (Fig.  216.)      The  number  of  teeth  on  a,  6,  c,  d,  e,  and  / 
are,  respectively,  15,  45,  23,  35,  1,  and  50.    Determine  velocity  ratio  be- 
tween axes  I  and  IV. 

59.  [Art.  135.]    (Fig.  217.)    The  cone-pulley  is  driven  by  an  equal  cone 
on  a  countershaft  which  makes  90  rev.  per  min.     The  steps  have  diame- 


256  KINEMATICS  OF  MACHINERY. 

ters  of  12",  9f,"  and  7".  The  gear  a  is  keyed  to  the  cone-pulley,  and  it  has 
28  teeth;  gears  6  and  c  are  fast  to  the  shaft  B,  and  have,  respectively,  100 
and  24  teeth;  d  is  keyed  to  the  spindle  and  has  88  teeth.  Calculate  the 
various  possible  speeds  of  the  spindle. 

60.  [Art,  137.]      The  lathe  has  a  lead  screw  with  4  threads  per  inch. 
The  change-gears  include  wheels  with  the  following  numbers  of  teeth:  24, 
30,  36,  42,  48,  48,  54,  60,  66,  69,  72,  78,  84.     The  "  stud  "  makes  the  same 
number  of  revolutions  as  the  spindle  in  a  given  time.     With  the  24-gear  on 
the  stud  what  gears  should  be  used  on  the  screw  to  cut  9,  10,  11,  11$  and  12 
threads,  respectively?   What  arrangement  would  be  used  to  cut  4  threads 
per  inch?    What  for  2  threads? 

61.  [Art.  137.J     Same  data  as  Prob.  60.     Arrange  table  showing  what 
gears  to  use  on  the  stud  and  screw  to  cut  threads  from  2  per  inch  up  to  14 
per  inch. 


INDEX. 


A 

PAGE 

Absolute  motion 3 

Acceleration 1 

Acceleration  diagrams 73 

Angular  velocity. . ... 18 

'*.  ratio 50,69 

"   ,  constant ! 53,56 

Angularity  of  connecting  rod 195 

Annular  wheels 122 

Approximate  tooth  profiles 136 

Axis,  instant '. 20 

Axodes...  75 


B 

Back-gears 238 

Backlash  and  clearance,  gears 129 

Bands 221 

Beam  motion , 210 

Bell-cranks 209 

Belt,  length  of 224 

Belts 221 

Belt-tighteners 230 

Bent  levers 209 

Bevel-gears 142 

"        '.*;.,  non-interchangeability  of 149 

"        "    ,  smoothness  in  operation  of 149 

Brush-wheels 107 

Burmester's  method  for  open  belts 228 

C 

Cams 169 

Cast  gears 157 

Centre,  instant 20,  60,  62,  66 

257 


258  INDEX. 

PAGE 

Centrodes 75 

Chain,  four-link 187 

Chain  wheels 230 

Change  gears 243 

Circular  pitch 129,  131 

Classes  of  gearing 110,  167 

Clearance  and  backlash,  gears 129 

Close-fitting  worm-wheel 183 

Common  methods  of  transmitting  motion 37 

Comparison  of  systems  of  gearing 128 

Composition  and  resolution  of  motion 14 

Condition  of  constant  angular  velocity  ratio 53 

positive  driving 57 

pure  rolling 54 

Cone  frictions 108 

Cone  pulleys 226,  239 

Cones,  rolling 91,  93 

Conjugate  teeth 112 

Connectors,  link 37 

,  wrapping 48,  221 

Constant  angular  velocity  ratio 53 

Constant  velocity  ratio  and  pure  rolling 56 

Constrained  motion 24 

Contact  transmission,  direct 37 

Continuous  motion 7,  189 

Corliss  wrist-plate  motion 210 

Crank  and  connecting-rod 192 

Crossed  belts 221,  225 

Crowning  pulleys 223 

Curvilinear  translation 9 

Cut  gears 158 

Cutters,  gear 162 

Cycle,  definition < 6 

Cylinder  cams 178 

Cylinders,  rolling 91 

Cycloids 116 

D 

Dead  points,  or  centres 187 

Describing  circles  in  gears 120 

Diagrams,  acceleration 73 

' '       ,  velocity 70 

Diametral  pitch 130 


INDEX.  259 

PAGE 

Dimensions  of  gear-teeth 131 

Direct-contact  transmission 37,  41 

Directional  relation  in  trains 238,  247 

Distance  of  centres  in  involute  gears 125 

Drag-link 189 

Driving,  positive 57 

E 

Eccentric 199 

Ellipses,  rolling 55,  80 

Epicyclic  trains 245 

Epicycloid 116 

Epicycloidal  system  of  gears 116 

teeth 117,  118 

Escapements •  •  •   220 

F 

Forces,  parallelogram  of 13 

Four-link  chain 187 

Free  motion 24 

Frictional  gearing 99 

G 

Gear-cutters 162 

Gearing,  tooth 110 

,  frictional 99 

Gear  moulding  machines 158 

Gear-planers 159,  164 

Clears,  bevel 142 

' '    ,  cast • 157 

"    ,  cut 158,  159,  161,  163 

' '    ,  helical 1 50 

' '    ,  non-circular 135 

11    ,  spiral 152 

' '    ,  stub  tooth 132 

"    ,  worm 180 

•Gear  teeth,  methods  of  cutting 158 

,  proportions  of 131 

Generating  circles,  epicycloidal  gears 120 

Grant's  odontographs 139 

Graphic  representation  of  motion 12 

Grooved  friction-wheels 101 

Guide-pulleys 230 


260  INDEX. 

H 

PAGES 

Helical  gears 150 

"        "    ,  graphical  method  for 154 

Helical  motion 7,  10 

Higher  pairing 38 

Hobbing  worm-wheels 184 

Hooke's  coupling 214 

Hyperboloids,  rolling 91,  97 

Hypocycloid 116 


I 

Idler  gear 240- 

Indicator  pencil  mechanisms 211 

Instant  axis. 2O 

Instant  centre 20,  60,  62,  66 

Instant  centre  theorem 64 

Interchangeable  set  of  gears 122. 

Interference  in  involute  gears 127 

Intermediate    connectors 37 

Intermittent  motion 7 

Inversion  of  mechanism 59,  204 

Involute  gearing 116,  124 

Involute  teeth,  interference  of ,. 127 


K 

Kennedy's  theorem 64 

Kinematics,  definition 34 


L 

Lazy-tongs 213 

Length  of  belts 224 

Length  of  connecting-rod 195 

Length  of  teeth 120 

link.?. 37 

Link-connectors 45 

LInkwork 186 

Lobcd  w-ieels 89 

Logarithmic  spirals,  rolling 55,  85 

Lower  pairing. 38- 


IXDEX.  261 

M 

PAGE 

Machine,  definition 29 

Machine  design,  definition, 34 

Mechanics,  definition 29 

Mechanism,  definition 29 

' '        ,  inversion  of 59 

1 '        ,  trains  of 233 

Methods  of  transmitting  motion 38 

Milling-cutters,  standard 158,  162 

"  bevel-gears 164 

1 '  spur-gears 161 

Mitre  gears 153 

Motion,  absolute 3 

' '     ,  continuous,  reciprocating  and  intermittent 7 

' '     ,  definition 1 

' '     ,  free  and  constrained 24 

' '     ,  graphic  representation  of 12 

"     ,  helical 10 

' '     ,  instantaneous. 20 

' '     ,  Newton's  laws  of 13 

' '     ,  plane . 7 

"     ,  relative 3,  61 

' '     ,  resolution  and  composition 14 

' '     ,  spherical 10 

N 

Newton's  laws  of  motion 13 

Non-circular  gears 135 

Non-interchangeability  of  bevel-gears 149 

O 

Obliquity  of  connecting-rod 195 

Open  belts 221,  225 

Oscillating-engine  mechanism 204 

Outlines  of  conjugate  gear-teeth 113 

Outlines  of  gear-teeth,  general  method 114 

helical  gear-teeth 153 

P 

Pairing,  higher  and  lower ; 38 

Pantographs .   213 


262  INDEX. 


_  ,,    ,  PAGE 

Parallel  motions 211 

Parallel  rods,  locomotive 190 

Parallelogram  of  forces 13 

' '  motions 15 

Path,  definition 6 

Pencil  motions,  indicator 211 

Period,  definition 5 

Phase,  definition 5 

Piston,  velocity  ratio  to  crank-pin 196 

Pitch  of  gear-teeth 129,  152 

surfaces 110,  135,  142,  151 

Planing  gear-teeth 159;  165 

Positive  driving  in  direct  contact 57 

Positive  return  cams 174 

Problems  and  exercises 249 

Proportions  of  gear-teeth 131 

Q 

Quick-return  motions 202,  204,  206 

Quarter-turn  belts 229 

R 

Rack  and  pinion 123 

Rapid  change  in  angular  motion  of  link 210 

Ratchets 217 

Rate  of  sliding  in  direct  contact 54 

Ratio,  velocity f .       5 

Reciprocating  motion 7 

Rectilinear  translation 9 

Relation  of  direction  of  rotation 52 

Relative  motion 3,  61 

Resolution  and  composition  of  motion 14 

Reverted  train 248 

Rolling  circles 79 

"      cones 91,  93 

' '      curves 78,  87 

' '      cylinders 91 

"     ellipses 55,  80 

"      hyperboloids 91,  97 

' '      logarithmic  spirals 55,  85 

' '      pure,  condition  of 54 

' '      surfaces .  .  90 


INDEX.  263 

PAGE 

Rolling  and  sliding 53 

Rope  transmission 221 

Rotation 7 

S 

Scotch  yoke , 201 

Screw 179 

Screw-cutting  train 242 

Shaper  quick-return  motion 206 

Sheaves  for  ropes 223 

Shifting  belts 223 

Side  rods,  locomotive 190 

Slider-crank  mechanism 60,  192 

Sliding,  rate  of 54 

Sliding  and  rolling 53 

Slip  in  frictional  gearing 106 

Spherical  motion 7,  10 

Spiral  gears 152 

Sprocket-wheels 231 

Stepped  cones 226 

Stepped  gearing- 133 

Straight-line  motions 211 

Strength  of  gear-teeth 129 

Stub  teeth 132 

Systems  of  gearing,  usual 116 

T 

Teeth,  conjugate 112 

' '    ,  epicycloidal 117 

' '    ,  involute ; 124 

"    ,  stepped 133 

"    ,  stub 132 

"    ,  twisted 133 

' '    ,  unsymmetrical 133 

' '    of  bevel  gears 145 

' '    of  gears,  proportions  of 131 

Tight-and-loose  pulleys 224 

Tooth-gearing 110 

Tooth  outlines,  general  methods 114 

Train,  screw-cutting 242 

Trains,  epicyclic 245 

Trains  of  mechanism. .  .  233 


264  INDEX. 

PAGE 

Translation  cams 177 

Translation,  rectilinear  and  curvilinear 7,  9 

Transmission  by  actual  contact 37 

without  material  connection 37 

TredgolcTs  approximate  method  for  bevel-gear  teeth 145 

Tumbling  gears 241 

Twisted  gearing 133 

U 

Universal  joint 214 

Unsymmetrical  teeth 133 

Unwin,  approximate  method  for  gear-teeth 136 

V 

' 'V"  friction-gears 101 

Value  of  a  train  of  mechanism 235 

Varying  angular  velocity,  wrapping  connectors 232 

Velocity,  angular , 18 

"     ,       ' "     ,  determined  by  instant  centres 69 

' '      diagrams 70,  71 

' '     ,  linear 1 

' '     ,       ' '  ,  determined  by  instant  centres 68 

11        ratio 5 

"           "  ,  helical  gears 153 

1 '     ,  uniform  and  variable 2 

W 

Wheels,  brush 107 

"     ,lobed 89 

Whitworth's  quick-return  mechanism 206 

Willis'  odontograph 137 

Worm  and  wheel 181 

Worm  gearing 180 

Wrapping  connectors ....• 48,  221 

Wrist-plate  motion 210 


